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Properties

Label 2-261-29.7-c1-0-10
Degree 22
Conductor 261261
Sign −0.823−0.567i-0.823 - 0.567i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + (−0.487 − 2.13i)2-s + (−2.52 + 1.21i)4-s + (−0.533 − 2.33i)5-s + (−1.21 − 0.586i)7-s + (1.09 + 1.37i)8-s + (−4.73 + 2.27i)10-s + (−2.18 + 2.74i)11-s + (3.76 − 4.71i)13-s + (−0.659 + 2.88i)14-s + (−1.08 + 1.36i)16-s − 3.05·17-s + (−5.07 + 2.44i)19-s + (4.19 + 5.25i)20-s + (6.93 + 3.33i)22-s + (0.225 − 0.989i)23-s + ⋯
L(s)  = 1  + (−0.344 − 1.51i)2-s + (−1.26 + 0.608i)4-s + (−0.238 − 1.04i)5-s + (−0.460 − 0.221i)7-s + (0.388 + 0.487i)8-s + (−1.49 + 0.720i)10-s + (−0.659 + 0.827i)11-s + (1.04 − 1.30i)13-s + (−0.176 + 0.772i)14-s + (−0.272 + 0.341i)16-s − 0.740·17-s + (−1.16 + 0.560i)19-s + (0.936 + 1.17i)20-s + (1.47 + 0.711i)22-s + (0.0470 − 0.206i)23-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((−0.823−0.567i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((−0.823−0.567i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: −0.823−0.567i-0.823 - 0.567i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(181,⋅)\chi_{261} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), −0.823−0.567i)(2,\ 261,\ (\ :1/2),\ -0.823 - 0.567i)

Particular Values

L(1)L(1) ≈\approx 0.204025+0.655631i0.204025 + 0.655631i
L(12)L(\frac12) ≈\approx 0.204025+0.655631i0.204025 + 0.655631i
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)
bad3 1 1
29 1+(−5.14+1.58i)T 1 + (-5.14 + 1.58i)T
good2 1+(0.487+2.13i)T+(−1.80+0.867i)T2 1 + (0.487 + 2.13i)T + (-1.80 + 0.867i)T^{2}
5 1+(0.533+2.33i)T+(−4.50+2.16i)T2 1 + (0.533 + 2.33i)T + (-4.50 + 2.16i)T^{2}
7 1+(1.21+0.586i)T+(4.36+5.47i)T2 1 + (1.21 + 0.586i)T + (4.36 + 5.47i)T^{2}
11 1+(2.18−2.74i)T+(−2.44−10.7i)T2 1 + (2.18 - 2.74i)T + (-2.44 - 10.7i)T^{2}
13 1+(−3.76+4.71i)T+(−2.89−12.6i)T2 1 + (-3.76 + 4.71i)T + (-2.89 - 12.6i)T^{2}
17 1+3.05T+17T2 1 + 3.05T + 17T^{2}
19 1+(5.07−2.44i)T+(11.8−14.8i)T2 1 + (5.07 - 2.44i)T + (11.8 - 14.8i)T^{2}
23 1+(−0.225+0.989i)T+(−20.7−9.97i)T2 1 + (-0.225 + 0.989i)T + (-20.7 - 9.97i)T^{2}
31 1+(1.80+7.90i)T+(−27.9+13.4i)T2 1 + (1.80 + 7.90i)T + (-27.9 + 13.4i)T^{2}
37 1+(−3.42−4.29i)T+(−8.23+36.0i)T2 1 + (-3.42 - 4.29i)T + (-8.23 + 36.0i)T^{2}
41 1−6.85T+41T2 1 - 6.85T + 41T^{2}
43 1+(−1.73+7.60i)T+(−38.7−18.6i)T2 1 + (-1.73 + 7.60i)T + (-38.7 - 18.6i)T^{2}
47 1+(−0.260+0.327i)T+(−10.4−45.8i)T2 1 + (-0.260 + 0.327i)T + (-10.4 - 45.8i)T^{2}
53 1+(2.96+13.0i)T+(−47.7+22.9i)T2 1 + (2.96 + 13.0i)T + (-47.7 + 22.9i)T^{2}
59 1+3.56T+59T2 1 + 3.56T + 59T^{2}
61 1+(6.49+3.12i)T+(38.0+47.6i)T2 1 + (6.49 + 3.12i)T + (38.0 + 47.6i)T^{2}
67 1+(−5.18−6.49i)T+(−14.9+65.3i)T2 1 + (-5.18 - 6.49i)T + (-14.9 + 65.3i)T^{2}
71 1+(−6.10+7.65i)T+(−15.7−69.2i)T2 1 + (-6.10 + 7.65i)T + (-15.7 - 69.2i)T^{2}
73 1+(−0.0675+0.296i)T+(−65.7−31.6i)T2 1 + (-0.0675 + 0.296i)T + (-65.7 - 31.6i)T^{2}
79 1+(−2.49−3.13i)T+(−17.5+77.0i)T2 1 + (-2.49 - 3.13i)T + (-17.5 + 77.0i)T^{2}
83 1+(3.88−1.87i)T+(51.7−64.8i)T2 1 + (3.88 - 1.87i)T + (51.7 - 64.8i)T^{2}
89 1+(−3.11−13.6i)T+(−80.1+38.6i)T2 1 + (-3.11 - 13.6i)T + (-80.1 + 38.6i)T^{2}
97 1+(−3.42+1.65i)T+(60.4−75.8i)T2 1 + (-3.42 + 1.65i)T + (60.4 - 75.8i)T^{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

−11.30840069998167300142922145052, −10.48970229861578215250622079591, −9.790938052111306607200394963826, −8.697177135044304119193021388268, −8.022009376939604986837135717876, −6.26511036816862802344597871968, −4.72585411475066579448688819284, −3.72237277089883871014844102693, −2.26625880041429465588239323201, −0.59186325370686096861902988804, 2.88875036758917725897457938881, 4.50915895419157384735886641107, 6.12492686338618327567345530772, 6.50178194147856312521672950622, 7.46346147748417644286253891964, 8.625960503925346173997671281528, 9.172428157614426484313916251818, 10.74411259979381560157363027696, 11.24198238850565782632621706874, 12.80550597669134438787494239361

Graph of the ZZ-function along the critical line