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Properties

Label 2-959-959.762-c0-0-0
Degree $2$
Conductor $959$
Sign $0.719 - 0.694i$
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  + (0.538 + 0.100i)2-s + (−0.653 − 0.253i)4-s + (−0.445 + 0.895i)7-s + (−0.791 − 0.489i)8-s + (0.273 + 0.961i)9-s + (1.83 + 0.710i)11-s + (−0.329 + 0.436i)14-s + (0.141 + 0.128i)16-s + (0.0505 + 0.544i)18-s + (0.914 + 0.566i)22-s + (0.907 − 0.995i)23-s + (−0.932 + 0.361i)25-s + (0.517 − 0.471i)28-s + (−0.247 + 0.271i)29-s + (0.623 + 0.826i)32-s + ⋯
L(s)  = 1  + (0.538 + 0.100i)2-s + (−0.653 − 0.253i)4-s + (−0.445 + 0.895i)7-s + (−0.791 − 0.489i)8-s + (0.273 + 0.961i)9-s + (1.83 + 0.710i)11-s + (−0.329 + 0.436i)14-s + (0.141 + 0.128i)16-s + (0.0505 + 0.544i)18-s + (0.914 + 0.566i)22-s + (0.907 − 0.995i)23-s + (−0.932 + 0.361i)25-s + (0.517 − 0.471i)28-s + (−0.247 + 0.271i)29-s + (0.623 + 0.826i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (762, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123123406\)
\(L(\frac12)\) \(\approx\) \(1.123123406\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.445 - 0.895i)T \)
137 \( 1 + T \)
good2 \( 1 + (-0.538 - 0.100i)T + (0.932 + 0.361i)T^{2} \)
3 \( 1 + (-0.273 - 0.961i)T^{2} \)
5 \( 1 + (0.932 - 0.361i)T^{2} \)
11 \( 1 + (-1.83 - 0.710i)T + (0.739 + 0.673i)T^{2} \)
13 \( 1 + (-0.602 - 0.798i)T^{2} \)
17 \( 1 + (-0.445 + 0.895i)T^{2} \)
19 \( 1 + (0.982 - 0.183i)T^{2} \)
23 \( 1 + (-0.907 + 0.995i)T + (-0.0922 - 0.995i)T^{2} \)
29 \( 1 + (0.247 - 0.271i)T + (-0.0922 - 0.995i)T^{2} \)
31 \( 1 + (-0.982 - 0.183i)T^{2} \)
37 \( 1 - 1.86T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.34 - 0.124i)T + (0.982 - 0.183i)T^{2} \)
47 \( 1 + (-0.850 + 0.526i)T^{2} \)
53 \( 1 + (1.58 - 0.147i)T + (0.982 - 0.183i)T^{2} \)
59 \( 1 + (0.850 - 0.526i)T^{2} \)
61 \( 1 + (0.850 + 0.526i)T^{2} \)
67 \( 1 + (1.72 + 0.857i)T + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (0.260 + 0.673i)T + (-0.739 + 0.673i)T^{2} \)
73 \( 1 + (0.602 - 0.798i)T^{2} \)
79 \( 1 + (-1.58 + 1.20i)T + (0.273 - 0.961i)T^{2} \)
83 \( 1 + (0.445 + 0.895i)T^{2} \)
89 \( 1 + (0.932 - 0.361i)T^{2} \)
97 \( 1 + (0.739 + 0.673i)T^{2} \)
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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

−10.12993691233596351738912083765, −9.361314564240078749907106646244, −8.960226975349904819128138675784, −7.83351866691175633155792692800, −6.63195847450644958929917123326, −6.05552521406225377092426313263, −4.93594416963542156866060928084, −4.32281225976137625198973571668, −3.18624161054447481165370853333, −1.73141058467629817108323076608, 1.08480369663458077543599622311, 3.27828717838248482434875538252, 3.79395098750250341240923658426, 4.52809847953318509541657695297, 5.94428472375527949521180188563, 6.51508538044770081953038526702, 7.53801774284854747245632970046, 8.634263200176548600625030276095, 9.453321726560069017954266590246, 9.772374556062796381805770663893

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