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Properties

Label 2-2151-2151.238-c0-0-9
Degree $2$
Conductor $2151$
Sign $0.438 - 0.898i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
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  + (0.374 + 0.648i)2-s + (0.669 + 0.743i)3-s + (0.219 − 0.379i)4-s + (−0.0348 + 0.0604i)5-s + (−0.231 + 0.712i)6-s + 1.07·8-s + (−0.104 + 0.994i)9-s − 0.0522·10-s + (−0.559 − 0.968i)11-s + (0.429 − 0.0912i)12-s + (−0.0682 + 0.0145i)15-s + (0.184 + 0.319i)16-s + 1.53·17-s + (−0.684 + 0.304i)18-s + (0.0153 + 0.0265i)20-s + ⋯
L(s)  = 1  + (0.374 + 0.648i)2-s + (0.669 + 0.743i)3-s + (0.219 − 0.379i)4-s + (−0.0348 + 0.0604i)5-s + (−0.231 + 0.712i)6-s + 1.07·8-s + (−0.104 + 0.994i)9-s − 0.0522·10-s + (−0.559 − 0.968i)11-s + (0.429 − 0.0912i)12-s + (−0.0682 + 0.0145i)15-s + (0.184 + 0.319i)16-s + 1.53·17-s + (−0.684 + 0.304i)18-s + (0.0153 + 0.0265i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.997528687\)
\(L(\frac12)\) \(\approx\) \(1.997528687\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.374 - 0.648i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.0348 - 0.0604i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.559 + 0.968i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.961 + 1.66i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.990 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.241 - 0.419i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.451291547740624135376252453785, −8.510888834134067215607145989138, −7.76395986949709003469001544551, −7.24863367801328790645964202196, −5.99078861679652900010806321729, −5.48039396875896108124065043356, −4.76273593210705614405684484171, −3.65528061960915712996399414038, −2.93625116568905334398190775322, −1.58535622799114614601859237730, 1.46544527216027558725802642510, 2.32072562679417954402229687527, 3.18034455314998352517255521099, 3.91931970285049054949398421792, 4.96816885337787099031157823156, 5.99532445171932532703775958661, 7.23314440961095321823994989699, 7.43895711313129437442033555473, 8.196461793389674003373036058166, 9.103317824831769205381417480916

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