Properties

Degree $2$
Conductor $261$
Sign $-0.999 + 0.0316i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 − 0.713i)2-s + (1.42 + 2.96i)4-s + (−4.21 + 0.962i)5-s + (−10.1 − 4.88i)7-s + (6.10 + 0.688i)8-s + (−1.20 + 3.44i)10-s + (−1.54 + 0.173i)11-s + (−11.3 − 9.02i)13-s + (−8.03 + 5.04i)14-s + (−4.97 + 6.23i)16-s + (−16.8 − 16.8i)17-s + (0.448 + 0.157i)19-s + (−8.87 − 11.1i)20-s + (−0.567 + 1.17i)22-s + (−1.55 + 6.79i)23-s + ⋯
L(s)  = 1  + (0.224 − 0.356i)2-s + (0.356 + 0.740i)4-s + (−0.843 + 0.192i)5-s + (−1.44 − 0.697i)7-s + (0.763 + 0.0860i)8-s + (−0.120 + 0.344i)10-s + (−0.140 + 0.0157i)11-s + (−0.870 − 0.694i)13-s + (−0.574 + 0.360i)14-s + (−0.310 + 0.389i)16-s + (−0.990 − 0.990i)17-s + (0.0236 + 0.00826i)19-s + (−0.443 − 0.556i)20-s + (−0.0258 + 0.0535i)22-s + (−0.0674 + 0.295i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.999 + 0.0316i$
Motivic weight: \(2\)
Character: $\chi_{261} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.999 + 0.0316i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00163347 - 0.103258i\)
\(L(\frac12)\) \(\approx\) \(0.00163347 - 0.103258i\)
\(L(2)\) 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 \( 1 + (-20.2 + 20.7i)T \)
good2 \( 1 + (-0.448 + 0.713i)T + (-1.73 - 3.60i)T^{2} \)
5 \( 1 + (4.21 - 0.962i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (10.1 + 4.88i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (1.54 - 0.173i)T + (117. - 26.9i)T^{2} \)
13 \( 1 + (11.3 + 9.02i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (16.8 + 16.8i)T + 289iT^{2} \)
19 \( 1 + (-0.448 - 0.157i)T + (282. + 225. i)T^{2} \)
23 \( 1 + (1.55 - 6.79i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (7.88 - 12.5i)T + (-416. - 865. i)T^{2} \)
37 \( 1 + (-26.2 - 2.96i)T + (1.33e3 + 304. i)T^{2} \)
41 \( 1 + (46.1 - 46.1i)T - 1.68e3iT^{2} \)
43 \( 1 + (53.7 - 33.7i)T + (802. - 1.66e3i)T^{2} \)
47 \( 1 + (5.72 + 50.7i)T + (-2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-9.09 - 39.8i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 23.8T + 3.48e3T^{2} \)
61 \( 1 + (-23.8 - 68.2i)T + (-2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (-89.8 + 71.6i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (-54.7 - 43.6i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (43.5 + 69.3i)T + (-2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-12.9 + 114. i)T + (-6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (-1.11 + 0.537i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-35.1 + 55.9i)T + (-3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (26.0 - 74.3i)T + (-7.35e3 - 5.86e3i)T^{2} \)
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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

−11.50217524232802247763543119402, −10.43086779883581478687164617951, −9.581439347398891632867333398375, −8.127996677186600387820602870584, −7.28238170097850917891100946560, −6.57897041351733859804487303773, −4.68561519958958597681093305088, −3.54997875044253149721530406791, −2.75259324492325218598885050533, −0.04502496751122291975323294420, 2.29775801122045559313475716094, 3.91152608030866874334785450177, 5.16523879006196061783136933097, 6.39023993425638215497368266432, 6.94933831071942727335832563309, 8.341469577078474402324558671750, 9.431729681376365395752759265590, 10.25112998649675164063054754460, 11.36029098269805924109785116750, 12.28350016216480319324750547897

Graph of the $Z$-function along the critical line