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Properties

Label 2-228-12.11-c1-0-7
Degree $2$
Conductor $228$
Sign $0.879 + 0.476i$
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 + (1.63 + 1.82i)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 + (1.65 − 3.04i)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 + (0.668 + 0.743i)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 + (0.476 − 0.879i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.879 + 0.476i$
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: \((2,\ 228,\ (\ :1/2),\ 0.879 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769799 - 0.195161i\)
\(L(\frac12)\) \(\approx\) \(0.769799 - 0.195161i\)
\(L(\frac{3}{2})\) 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

−12.04535765248717317813596559286, −10.99318801456669803166302050532, −10.61222006958280298255445083361, −9.251896623453859043164950837949, −8.724242863553672346765994493930, −7.06438038202703635654049990225, −5.80026813566702498181947361916, −4.52679797963646835777321516947, −3.44394989685410910375825219434, −1.28125149783628628032595739781, 1.12767701134980973463576558141, 4.07663189997252852320034619886, 5.31106229746218515436430058283, 6.38844222934476913406470044144, 6.99994256820501244196536075058, 8.187113826780321927231354307404, 9.250545279358042928718564338779, 10.40709194077005974897188473689, 11.20003539337803576015970326986, 12.31816571706316853576854705242

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