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Properties

Label 2-228-12.11-c1-0-33
Degree $2$
Conductor $228$
Sign $-0.290 + 0.956i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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  + (0.474 − 1.33i)2-s + (1.60 − 0.658i)3-s + (−1.54 − 1.26i)4-s − 0.191i·5-s + (−0.116 − 2.44i)6-s − 0.733i·7-s + (−2.41 + 1.46i)8-s + (2.13 − 2.10i)9-s + (−0.255 − 0.0910i)10-s − 2.95·11-s + (−3.31 − 1.00i)12-s + 4.31·13-s + (−0.977 − 0.348i)14-s + (−0.126 − 0.307i)15-s + (0.802 + 3.91i)16-s + 1.98i·17-s + ⋯
L(s)  = 1  + (0.335 − 0.942i)2-s + (0.924 − 0.379i)3-s + (−0.774 − 0.632i)4-s − 0.0857i·5-s + (−0.0475 − 0.998i)6-s − 0.277i·7-s + (−0.855 + 0.517i)8-s + (0.711 − 0.702i)9-s + (−0.0808 − 0.0287i)10-s − 0.891·11-s + (−0.956 − 0.290i)12-s + 1.19·13-s + (−0.261 − 0.0930i)14-s + (−0.0326 − 0.0793i)15-s + (0.200 + 0.979i)16-s + 0.481i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02699 - 1.38490i\)
\(L(\frac12)\) \(\approx\) \(1.02699 - 1.38490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.474 + 1.33i)T \)
3 \( 1 + (-1.60 + 0.658i)T \)
19 \( 1 - iT \)
good5 \( 1 + 0.191iT - 5T^{2} \)
7 \( 1 + 0.733iT - 7T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 - 1.98iT - 17T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 - 8.26iT - 29T^{2} \)
31 \( 1 - 4.66iT - 31T^{2} \)
37 \( 1 + 5.08T + 37T^{2} \)
41 \( 1 + 6.49iT - 41T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 + 6.14T + 47T^{2} \)
53 \( 1 - 7.35iT - 53T^{2} \)
59 \( 1 - 9.63T + 59T^{2} \)
61 \( 1 + 1.98T + 61T^{2} \)
67 \( 1 + 0.0795iT - 67T^{2} \)
71 \( 1 + 8.52T + 71T^{2} \)
73 \( 1 + 5.26T + 73T^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 - 0.0511T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−12.23122334328957464396899918946, −10.80603213963432935609163937716, −10.25693285174306694989443834541, −8.898871345366003621422904990900, −8.367372486397236108982196902281, −6.92726910436287735363769066592, −5.48455166672264755441050643104, −4.01547688137618759979793324941, −3.01630626190077579406366852757, −1.50455020300490086504067496198, 2.78739281109130223670597736834, 4.03258817219784711603010282280, 5.19517948186230683657244127707, 6.43638714390249742905388991394, 7.71987372770223469062886032834, 8.381999943490089993646925119758, 9.305200873304826996044280709490, 10.31710448851541041812154984194, 11.65538417593760157663076635605, 13.13848108689244898502028661768

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