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Properties

Label 1-2557-2557.78-r1-0-0
Degree $1$
Conductor $2557$
Sign $-0.297 - 0.954i$
Analytic cond. $274.787$
Root an. cond. $274.787$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $-0.297 - 0.954i$
Analytic conductor: \(274.787\)
Root analytic conductor: \(274.787\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (1:\ ),\ -0.297 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8044086381 - 1.093077132i\)
\(L(\frac12)\) \(\approx\) \(0.8044086381 - 1.093077132i\)
\(L(1)\) \(\approx\) \(0.5873803487 - 0.4081111737i\)
\(L(1)\) \(\approx\) \(0.5873803487 - 0.4081111737i\)

Euler product

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

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 $Z$-function along the critical line