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Properties

Label 2-1183-7.4-c1-0-83
Degree 22
Conductor 11831183
Sign −0.902+0.431i-0.902 + 0.431i
Analytic cond. 9.446309.44630
Root an. cond. 3.073483.07348
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.00118 − 0.00205i)2-s + (1.54 − 2.68i)3-s + (0.999 − 1.73i)4-s + (0.565 + 0.979i)5-s − 0.00733·6-s + (−2.64 − 0.0383i)7-s − 0.00946·8-s + (−3.29 − 5.71i)9-s + (0.00133 − 0.00231i)10-s + (2.23 − 3.87i)11-s + (−3.09 − 5.36i)12-s + (0.00305 + 0.00546i)14-s + 3.50·15-s + (−1.99 − 3.46i)16-s + (−1.94 + 3.36i)17-s + (−0.00781 + 0.0135i)18-s + ⋯
L(s)  = 1  + (−0.000836 − 0.00144i)2-s + (0.894 − 1.54i)3-s + (0.499 − 0.866i)4-s + (0.252 + 0.438i)5-s − 0.00299·6-s + (−0.999 − 0.0144i)7-s − 0.00334·8-s + (−1.09 − 1.90i)9-s + (0.000423 − 0.000733i)10-s + (0.675 − 1.16i)11-s + (−0.894 − 1.54i)12-s + (0.000815 + 0.00146i)14-s + 0.905·15-s + (−0.499 − 0.866i)16-s + (−0.471 + 0.816i)17-s + (−0.00184 + 0.00318i)18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((−0.902+0.431i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((−0.902+0.431i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: −0.902+0.431i-0.902 + 0.431i
Analytic conductor: 9.446309.44630
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1183(508,⋅)\chi_{1183} (508, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1183, ( :1/2), −0.902+0.431i)(2,\ 1183,\ (\ :1/2),\ -0.902 + 0.431i)

Particular Values

L(1)L(1) ≈\approx 2.1742246922.174224692
L(12)L(\frac12) ≈\approx 2.1742246922.174224692
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1+(2.64+0.0383i)T 1 + (2.64 + 0.0383i)T
13 1 1
good2 1+(0.00118+0.00205i)T+(−1+1.73i)T2 1 + (0.00118 + 0.00205i)T + (-1 + 1.73i)T^{2}
3 1+(−1.54+2.68i)T+(−1.5−2.59i)T2 1 + (-1.54 + 2.68i)T + (-1.5 - 2.59i)T^{2}
5 1+(−0.565−0.979i)T+(−2.5+4.33i)T2 1 + (-0.565 - 0.979i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.23+3.87i)T+(−5.5−9.52i)T2 1 + (-2.23 + 3.87i)T + (-5.5 - 9.52i)T^{2}
17 1+(1.94−3.36i)T+(−8.5−14.7i)T2 1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.27−2.20i)T+(−9.5+16.4i)T2 1 + (-1.27 - 2.20i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.459−0.796i)T+(−11.5+19.9i)T2 1 + (-0.459 - 0.796i)T + (-11.5 + 19.9i)T^{2}
29 1+4.32T+29T2 1 + 4.32T + 29T^{2}
31 1+(3.12−5.40i)T+(−15.5−26.8i)T2 1 + (3.12 - 5.40i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.769−1.33i)T+(−18.5+32.0i)T2 1 + (-0.769 - 1.33i)T + (-18.5 + 32.0i)T^{2}
41 1−6.86T+41T2 1 - 6.86T + 41T^{2}
43 1−12.1T+43T2 1 - 12.1T + 43T^{2}
47 1+(4.71+8.16i)T+(−23.5+40.7i)T2 1 + (4.71 + 8.16i)T + (-23.5 + 40.7i)T^{2}
53 1+(−4.91+8.50i)T+(−26.5−45.8i)T2 1 + (-4.91 + 8.50i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.87−8.44i)T+(−29.5−51.0i)T2 1 + (4.87 - 8.44i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.21−3.83i)T+(−30.5+52.8i)T2 1 + (-2.21 - 3.83i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.44+2.51i)T+(−33.5−58.0i)T2 1 + (-1.44 + 2.51i)T + (-33.5 - 58.0i)T^{2}
71 1+11.5T+71T2 1 + 11.5T + 71T^{2}
73 1+(−6.53+11.3i)T+(−36.5−63.2i)T2 1 + (-6.53 + 11.3i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.0197+0.0342i)T+(−39.5+68.4i)T2 1 + (0.0197 + 0.0342i)T + (-39.5 + 68.4i)T^{2}
83 1+4.33T+83T2 1 + 4.33T + 83T^{2}
89 1+(−1.83−3.17i)T+(−44.5+77.0i)T2 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2}
97 1+2.12T+97T2 1 + 2.12T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.170179449738901997681382627988, −8.702607874232726275320193356403, −7.55027197568447934120668644605, −6.85059659585082968955318393594, −6.21030438996862032313036865841, −5.79795344942637166252553150682, −3.68797530300745632437919567222, −2.84615397799948648577204659570, −1.91301026191852800913886414668, −0.811604755713985767507097402966, 2.29749841157763536997257406704, 3.05366808498089254534224420411, 4.03198578502505398066173952959, 4.55588828469166065177034096996, 5.79237125715652970385064083256, 7.05301760271161118393187971848, 7.67284477789376010689184247971, 8.946244061510756272096621826503, 9.289405675155825974485130174959, 9.709692874828528798960046156062

Graph of the ZZ-function along the critical line