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Properties

Label 1-2557-2557.78-r1-0-0
Degree 11
Conductor 25572557
Sign −0.297−0.954i-0.297 - 0.954i
Analytic cond. 274.787274.787
Root an. cond. 274.787274.787
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.337 − 0.941i)2-s + (−0.637 − 0.770i)3-s + (−0.772 + 0.635i)4-s + (−0.741 + 0.670i)5-s + (−0.510 + 0.859i)6-s + (−0.570 − 0.821i)7-s + (0.858 + 0.512i)8-s + (−0.188 + 0.982i)9-s + (0.881 + 0.472i)10-s + (0.999 − 0.0393i)11-s + (0.981 + 0.190i)12-s + (0.867 + 0.497i)13-s + (−0.580 + 0.814i)14-s + (0.989 + 0.144i)15-s + (0.192 − 0.981i)16-s + (0.947 + 0.321i)17-s + ⋯
L(s)  = 1  + (−0.337 − 0.941i)2-s + (−0.637 − 0.770i)3-s + (−0.772 + 0.635i)4-s + (−0.741 + 0.670i)5-s + (−0.510 + 0.859i)6-s + (−0.570 − 0.821i)7-s + (0.858 + 0.512i)8-s + (−0.188 + 0.982i)9-s + (0.881 + 0.472i)10-s + (0.999 − 0.0393i)11-s + (0.981 + 0.190i)12-s + (0.867 + 0.497i)13-s + (−0.580 + 0.814i)14-s + (0.989 + 0.144i)15-s + (0.192 − 0.981i)16-s + (0.947 + 0.321i)17-s + ⋯

Functional equation

Λ(s)=(2557s/2 ΓR(s+1) L(s)=((−0.297−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2557s/2 ΓR(s+1) L(s)=((−0.297−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 25572557
Sign: −0.297−0.954i-0.297 - 0.954i
Analytic conductor: 274.787274.787
Root analytic conductor: 274.787274.787
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2557(78,⋅)\chi_{2557} (78, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2557, (1: ), −0.297−0.954i)(1,\ 2557,\ (1:\ ),\ -0.297 - 0.954i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.8044086381−1.093077132i0.8044086381 - 1.093077132i
L(12)L(\frac12) ≈\approx 0.8044086381−1.093077132i0.8044086381 - 1.093077132i
L(1)L(1) ≈\approx 0.5873803487−0.4081111737i0.5873803487 - 0.4081111737i
L(1)L(1) ≈\approx 0.5873803487−0.4081111737i0.5873803487 - 0.4081111737i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2557 1 1
good2 1+(−0.337−0.941i)T 1 + (-0.337 - 0.941i)T
3 1+(−0.637−0.770i)T 1 + (-0.637 - 0.770i)T
5 1+(−0.741+0.670i)T 1 + (-0.741 + 0.670i)T
7 1+(−0.570−0.821i)T 1 + (-0.570 - 0.821i)T
11 1+(0.999−0.0393i)T 1 + (0.999 - 0.0393i)T
13 1+(0.867+0.497i)T 1 + (0.867 + 0.497i)T
17 1+(0.947+0.321i)T 1 + (0.947 + 0.321i)T
19 1+(0.202−0.979i)T 1 + (0.202 - 0.979i)T
23 1+(−0.578+0.815i)T 1 + (-0.578 + 0.815i)T
29 1+(0.330−0.943i)T 1 + (0.330 - 0.943i)T
31 1+(0.994−0.105i)T 1 + (0.994 - 0.105i)T
37 1+(0.817−0.576i)T 1 + (0.817 - 0.576i)T
41 1+(0.560−0.828i)T 1 + (0.560 - 0.828i)T
43 1+(0.916−0.399i)T 1 + (0.916 - 0.399i)T
47 1+(−0.497−0.867i)T 1 + (-0.497 - 0.867i)T
53 1+(0.525+0.850i)T 1 + (0.525 + 0.850i)T
59 1+(−0.529+0.848i)T 1 + (-0.529 + 0.848i)T
61 1+(−0.787+0.616i)T 1 + (-0.787 + 0.616i)T
67 1+(0.994+0.100i)T 1 + (0.994 + 0.100i)T
71 1+(0.478−0.878i)T 1 + (0.478 - 0.878i)T
73 1+(0.541+0.840i)T 1 + (0.541 + 0.840i)T
79 1+(0.467−0.883i)T 1 + (0.467 - 0.883i)T
83 1+(0.295−0.955i)T 1 + (0.295 - 0.955i)T
89 1+(0.882−0.469i)T 1 + (0.882 - 0.469i)T
97 1+(0.0515−0.998i)T 1 + (0.0515 - 0.998i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.318561408715562783555390671401, −18.51626709653053259590660682462, −17.97912388872301325495531069088, −16.98991505592642800623287928890, −16.39309632664700342020730363920, −16.09775013320268884716426141825, −15.445390913506892141269533132244, −14.70551430202397174939826498670, −14.08983438059346401562160408842, −12.7521000563024716639022655803, −12.333532237971758734732745673802, −11.53878688407728663623694648879, −10.647722665717380994782329585327, −9.63835991767327475893467354308, −9.390054399849074891214209333276, −8.36231895438381776389557713763, −8.01826286557689454791960472633, −6.62384191603226914003760670707, −6.16472871895847928520286539656, −5.43598285502016130934306280376, −4.685993400922121703083313144494, −3.88836768186830788154181758096, −3.20614587153284766505996563952, −1.24813576066025755995709667785, −0.65520208909955017303887440372, 0.59449746557562890739644874166, 0.99198484840582949565740950956, 2.11873145322191264144981261101, 3.11698221735581985385223126793, 3.91032981573309891985773147062, 4.41877371699100985806416318771, 5.85703962798614539929317739654, 6.571961612806886154235871642334, 7.40163970392398614794044998945, 7.85763236486868591710759374557, 8.87879545046304527582977118947, 9.7715661260712343251862999359, 10.60602625257569761605997429860, 11.1027913360583934575365194960, 11.907195112497814734037969683598, 12.1247635149052702832679091894, 13.304269599456432358860675376228, 13.72790313993820654866594192877, 14.38028002378510694255540667763, 15.7090231758426002854030595912, 16.40587721085373372020223353146, 17.16002229995333477480257079183, 17.64660331844894337718649335087, 18.56686550398657649004035841405, 19.030156062123462639269197278

Graph of the ZZ-function along the critical line