Properties

Degree $2$
Conductor $261$
Sign $0.934 - 0.357i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 + 0.907i)2-s + (−0.472 + 0.980i)4-s + (8.15 + 1.86i)5-s + (7.53 − 3.62i)7-s + (−0.972 − 8.62i)8-s + (−13.4 + 4.71i)10-s + (1.08 − 9.61i)11-s + (8.88 − 7.08i)13-s + (−7.59 + 12.0i)14-s + (6.52 + 8.17i)16-s + (−11.2 − 11.2i)17-s + (−5.31 − 15.1i)19-s + (−5.67 + 7.12i)20-s + (7.16 + 14.8i)22-s + (2.98 + 13.0i)23-s + ⋯
L(s)  = 1  + (−0.722 + 0.453i)2-s + (−0.118 + 0.245i)4-s + (1.63 + 0.372i)5-s + (1.07 − 0.518i)7-s + (−0.121 − 1.07i)8-s + (−1.34 + 0.471i)10-s + (0.0985 − 0.874i)11-s + (0.683 − 0.544i)13-s + (−0.542 + 0.863i)14-s + (0.407 + 0.511i)16-s + (−0.664 − 0.664i)17-s + (−0.279 − 0.799i)19-s + (−0.283 + 0.356i)20-s + (0.325 + 0.676i)22-s + (0.129 + 0.569i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.357i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.934 - 0.357i$
Motivic weight: \(2\)
Character: $\chi_{261} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.934 - 0.357i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48641 + 0.274547i\)
\(L(\frac12)\) \(\approx\) \(1.48641 + 0.274547i\)
\(L(2)\) 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 \( 1 + (-25.9 + 13.0i)T \)
good2 \( 1 + (1.44 - 0.907i)T + (1.73 - 3.60i)T^{2} \)
5 \( 1 + (-8.15 - 1.86i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (-7.53 + 3.62i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-1.08 + 9.61i)T + (-117. - 26.9i)T^{2} \)
13 \( 1 + (-8.88 + 7.08i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (11.2 + 11.2i)T + 289iT^{2} \)
19 \( 1 + (5.31 + 15.1i)T + (-282. + 225. i)T^{2} \)
23 \( 1 + (-2.98 - 13.0i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (-1.32 + 0.832i)T + (416. - 865. i)T^{2} \)
37 \( 1 + (-2.36 - 21.0i)T + (-1.33e3 + 304. i)T^{2} \)
41 \( 1 + (-0.193 + 0.193i)T - 1.68e3iT^{2} \)
43 \( 1 + (38.8 - 61.7i)T + (-802. - 1.66e3i)T^{2} \)
47 \( 1 + (42.6 + 4.80i)T + (2.15e3 + 491. i)T^{2} \)
53 \( 1 + (17.1 - 75.1i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 23.4T + 3.48e3T^{2} \)
61 \( 1 + (-56.8 - 19.9i)T + (2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (48.7 + 38.9i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (79.7 - 63.6i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (23.3 + 14.6i)T + (2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-11.7 + 1.32i)T + (6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (0.116 + 0.0561i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (24.8 - 15.6i)T + (3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (-23.5 + 8.24i)T + (7.35e3 - 5.86e3i)T^{2} \)
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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

−11.50488370839278211492536077917, −10.67725413246480287172352067439, −9.763067555207455727454345260742, −8.860694211558930999181561291664, −8.070277745288912829228309429586, −6.87423373638097111186208797243, −6.01519207770912580533714641548, −4.69193318107794937463130251729, −2.95140409096722513258704716444, −1.18717319806482994445919381717, 1.61198215555362390406837778706, 2.06464083273635425053291212946, 4.65726151382853360983168098620, 5.53815999033473641812127515666, 6.54126824610824125897395987189, 8.425439344827548566374898322109, 8.828364714475051633750450986651, 9.874548687469442589890990426798, 10.47715569142436932122781154812, 11.47029443065696212001096045086

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