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Properties

Label 2-2156-308.207-c0-0-1
Degree $2$
Conductor $2156$
Sign $0.440 - 0.897i$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)18-s + (−0.809 + 0.587i)22-s + (1.01 − 0.587i)23-s + (0.104 + 0.994i)25-s + (0.5 + 0.363i)29-s + (0.500 − 0.866i)32-s + (0.309 + 0.951i)36-s + (−0.169 + 1.60i)37-s − 1.90i·43-s + (0.104 − 0.994i)44-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)18-s + (−0.809 + 0.587i)22-s + (1.01 − 0.587i)23-s + (0.104 + 0.994i)25-s + (0.5 + 0.363i)29-s + (0.500 − 0.866i)32-s + (0.309 + 0.951i)36-s + (−0.169 + 1.60i)37-s − 1.90i·43-s + (0.104 − 0.994i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ 0.440 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8069723995\)
\(L(\frac12)\) \(\approx\) \(0.8069723995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 \)
11 \( 1 + (-0.978 - 0.207i)T \)
good3 \( 1 + (0.978 - 0.207i)T^{2} \)
5 \( 1 + (-0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.913 + 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (-1.01 + 0.587i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.104 - 0.994i)T^{2} \)
37 \( 1 + (0.169 - 1.60i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.90iT - T^{2} \)
47 \( 1 + (-0.669 - 0.743i)T^{2} \)
53 \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \)
59 \( 1 + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.104 - 0.994i)T^{2} \)
67 \( 1 + (-1.64 - 0.951i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.669 - 0.743i)T^{2} \)
79 \( 1 + (-0.395 - 1.86i)T + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−9.130028038160275433510137426693, −8.685942677687999537050725162291, −7.974482428991566736845477576649, −6.92055649974477032959923967532, −6.56991300137705390186050617789, −5.48219773625376493759171818070, −4.93703105365822843540017382220, −3.71095434632529169431390736870, −2.45755004255016039316357905292, −1.14768862277056526561571673262, 0.880608936908200303754094222642, 2.23007909578011927921338333246, 3.17529608929992371105153780895, 3.95166811278352345061858988313, 4.99512658484538926689602366344, 6.15128604865641851477438993031, 6.85764378650936362607384150821, 7.86902691816112529357822568261, 8.514509314307978002327384301147, 9.218746893590908631462904600639

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