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Properties

Label 2-261-29.7-c1-0-10
Degree $2$
Conductor $261$
Sign $-0.823 - 0.567i$
Analytic cond. $2.08409$
Root an. cond. $1.44363$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ -0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204025 + 0.655631i\)
\(L(\frac12)\) \(\approx\) \(0.204025 + 0.655631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line