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Properties

Label 2-959-959.629-c0-0-0
Degree $2$
Conductor $959$
Sign $-0.941 - 0.335i$
Analytic cond. $0.478603$
Root an. cond. $0.691811$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 1.56i)2-s + (−0.784 + 2.75i)4-s + (−0.739 − 0.673i)7-s + (−3.41 + 1.32i)8-s + (0.982 + 0.183i)9-s + (−0.329 + 1.15i)11-s + (0.181 − 1.95i)14-s + (−3.69 − 2.28i)16-s + (0.876 + 1.75i)18-s + (−2.20 + 0.855i)22-s + (0.554 − 0.895i)23-s + (0.273 + 0.961i)25-s + (2.43 − 1.50i)28-s + (0.840 − 1.35i)29-s + (−0.449 − 4.85i)32-s + ⋯
L(s)  = 1  + (1.18 + 1.56i)2-s + (−0.784 + 2.75i)4-s + (−0.739 − 0.673i)7-s + (−3.41 + 1.32i)8-s + (0.982 + 0.183i)9-s + (−0.329 + 1.15i)11-s + (0.181 − 1.95i)14-s + (−3.69 − 2.28i)16-s + (0.876 + 1.75i)18-s + (−2.20 + 0.855i)22-s + (0.554 − 0.895i)23-s + (0.273 + 0.961i)25-s + (2.43 − 1.50i)28-s + (0.840 − 1.35i)29-s + (−0.449 − 4.85i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $-0.941 - 0.335i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ -0.941 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.600367279\)
\(L(\frac12)\) \(\approx\) \(1.600367279\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.739 + 0.673i)T \)
137 \( 1 + T \)
good2 \( 1 + (-1.18 - 1.56i)T + (-0.273 + 0.961i)T^{2} \)
3 \( 1 + (-0.982 - 0.183i)T^{2} \)
5 \( 1 + (-0.273 - 0.961i)T^{2} \)
11 \( 1 + (0.329 - 1.15i)T + (-0.850 - 0.526i)T^{2} \)
13 \( 1 + (0.0922 + 0.995i)T^{2} \)
17 \( 1 + (-0.739 - 0.673i)T^{2} \)
19 \( 1 + (0.602 - 0.798i)T^{2} \)
23 \( 1 + (-0.554 + 0.895i)T + (-0.445 - 0.895i)T^{2} \)
29 \( 1 + (-0.840 + 1.35i)T + (-0.445 - 0.895i)T^{2} \)
31 \( 1 + (-0.602 - 0.798i)T^{2} \)
37 \( 1 + 0.547T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.942 + 0.469i)T + (0.602 - 0.798i)T^{2} \)
47 \( 1 + (0.932 + 0.361i)T^{2} \)
53 \( 1 + (-1.78 + 0.887i)T + (0.602 - 0.798i)T^{2} \)
59 \( 1 + (-0.932 - 0.361i)T^{2} \)
61 \( 1 + (-0.932 + 0.361i)T^{2} \)
67 \( 1 + (-0.247 + 0.271i)T + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (1.85 - 0.526i)T + (0.850 - 0.526i)T^{2} \)
73 \( 1 + (-0.0922 + 0.995i)T^{2} \)
79 \( 1 + (1.78 - 0.165i)T + (0.982 - 0.183i)T^{2} \)
83 \( 1 + (0.739 - 0.673i)T^{2} \)
89 \( 1 + (-0.273 - 0.961i)T^{2} \)
97 \( 1 + (-0.850 - 0.526i)T^{2} \)
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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

−10.44759594403151209719916219401, −9.615543701238869758676215606413, −8.613552062974365661165053993467, −7.53865090202996203033371097460, −7.12011225425729024824662742964, −6.50588437669573245337533726564, −5.40233165378491849750829958139, −4.47810013661094383097164801233, −3.96951746357851652311737917320, −2.69329381089355561226914867858, 1.20292350742541234857455084095, 2.64412711623360736253359120611, 3.32114810962724591258261691113, 4.28336521249946142206355259093, 5.31239504944118468909229477779, 6.00567864870683134668113644698, 6.92617693260679231790264518074, 8.708497023924512351631232180312, 9.312022488488002682655648884018, 10.27973136976336156599556150816

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