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Properties

Label 2-837-279.254-c1-0-18
Degree $2$
Conductor $837$
Sign $0.964 - 0.262i$
Analytic cond. $6.68347$
Root an. cond. $2.58524$
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  + (1.45 + 0.840i)2-s + (0.411 + 0.712i)4-s + 0.175i·5-s + 4.05·7-s − 1.97i·8-s + (−0.147 + 0.254i)10-s + (−0.246 − 0.426i)11-s − 2.38i·13-s + (5.90 + 3.40i)14-s + (2.48 − 4.30i)16-s + (0.0216 + 0.0375i)17-s + (−1.32 − 2.29i)19-s + (−0.124 + 0.0720i)20-s − 0.826i·22-s + (1.81 + 3.13i)23-s + ⋯
L(s)  = 1  + (1.02 + 0.593i)2-s + (0.205 + 0.356i)4-s + 0.0783i·5-s + 1.53·7-s − 0.699i·8-s + (−0.0465 + 0.0806i)10-s + (−0.0741 − 0.128i)11-s − 0.662i·13-s + (1.57 + 0.910i)14-s + (0.621 − 1.07i)16-s + (0.00526 + 0.00911i)17-s + (−0.304 − 0.527i)19-s + (−0.0279 + 0.0161i)20-s − 0.176i·22-s + (0.377 + 0.653i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.964 - 0.262i$
Analytic conductor: \(6.68347\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :1/2),\ 0.964 - 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96418 + 0.395727i\)
\(L(\frac12)\) \(\approx\) \(2.96418 + 0.395727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-2.67 - 4.88i)T \)
good2 \( 1 + (-1.45 - 0.840i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 0.175iT - 5T^{2} \)
7 \( 1 - 4.05T + 7T^{2} \)
11 \( 1 + (0.246 + 0.426i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.38iT - 13T^{2} \)
17 \( 1 + (-0.0216 - 0.0375i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 + 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 3.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.85 - 6.68i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.97 - 2.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.87iT - 41T^{2} \)
43 \( 1 + 9.09iT - 43T^{2} \)
47 \( 1 + (-5.61 - 3.24i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.09 - 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.67 + 5.01i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.20 - 4.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + 3.24T + 67T^{2} \)
71 \( 1 + (-1.70 - 0.984i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.33 - 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.529iT - 79T^{2} \)
83 \( 1 + (-3.43 - 5.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.98T + 89T^{2} \)
97 \( 1 + (-1.29 + 2.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60111795604542780389665639197, −9.229295568860631547091145260144, −8.407115283384476597861917016623, −7.43126765623256968501520147884, −6.77275030084660273965747532816, −5.44210289334540294817565248581, −5.13925962794548993442578206276, −4.16447747771094678694133437359, −2.99744341886182112627649060565, −1.32191943939628889062645974591, 1.63777190990323218157886057333, 2.59575353382779394954313630230, 4.01027367568495211458887984377, 4.62623620571514452081978423961, 5.37351906418313852600914812857, 6.48452051334462950728007781407, 7.82893856265314427945107524560, 8.310103405698243550527095007156, 9.346070410667230221375754682953, 10.58216144286278673034794111819

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