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Properties

Label 2-2008-2008.179-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.998 - 0.0618i$
Analytic cond. $1.00212$
Root an. cond. $1.00106$
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  + (−0.637 − 0.770i)2-s + (0.238 + 0.138i)3-s + (−0.187 + 0.982i)4-s + (−0.0448 − 0.271i)6-s + (0.876 − 0.481i)8-s + (−0.455 − 0.804i)9-s + (−0.139 + 1.58i)11-s + (−0.180 + 0.207i)12-s + (−0.929 − 0.368i)16-s + (1.43 + 0.743i)17-s + (−0.329 + 0.863i)18-s + (−0.125 − 0.00945i)19-s + (1.30 − 0.900i)22-s + (0.275 + 0.00692i)24-s + (0.0627 + 0.998i)25-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)2-s + (0.238 + 0.138i)3-s + (−0.187 + 0.982i)4-s + (−0.0448 − 0.271i)6-s + (0.876 − 0.481i)8-s + (−0.455 − 0.804i)9-s + (−0.139 + 1.58i)11-s + (−0.180 + 0.207i)12-s + (−0.929 − 0.368i)16-s + (1.43 + 0.743i)17-s + (−0.329 + 0.863i)18-s + (−0.125 − 0.00945i)19-s + (1.30 − 0.900i)22-s + (0.275 + 0.00692i)24-s + (0.0627 + 0.998i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.998 - 0.0618i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.998 - 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8578252693\)
\(L(\frac12)\) \(\approx\) \(0.8578252693\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.637 + 0.770i)T \)
251 \( 1 + (0.947 - 0.320i)T \)
good3 \( 1 + (-0.238 - 0.138i)T + (0.492 + 0.870i)T^{2} \)
5 \( 1 + (-0.0627 - 0.998i)T^{2} \)
7 \( 1 + (-0.793 - 0.607i)T^{2} \)
11 \( 1 + (0.139 - 1.58i)T + (-0.984 - 0.175i)T^{2} \)
13 \( 1 + (0.974 + 0.224i)T^{2} \)
17 \( 1 + (-1.43 - 0.743i)T + (0.577 + 0.816i)T^{2} \)
19 \( 1 + (0.125 + 0.00945i)T + (0.988 + 0.150i)T^{2} \)
23 \( 1 + (-0.0125 + 0.999i)T^{2} \)
29 \( 1 + (-0.402 - 0.915i)T^{2} \)
31 \( 1 + (0.470 - 0.882i)T^{2} \)
37 \( 1 + (0.0878 + 0.996i)T^{2} \)
41 \( 1 + (-0.917 + 0.702i)T + (0.260 - 0.965i)T^{2} \)
43 \( 1 + (0.422 - 0.791i)T + (-0.556 - 0.830i)T^{2} \)
47 \( 1 + (-0.728 + 0.684i)T^{2} \)
53 \( 1 + (-0.617 - 0.786i)T^{2} \)
59 \( 1 + (-0.913 + 0.472i)T + (0.577 - 0.816i)T^{2} \)
61 \( 1 + (0.962 + 0.272i)T^{2} \)
67 \( 1 + (0.434 - 1.46i)T + (-0.837 - 0.546i)T^{2} \)
71 \( 1 + (-0.656 - 0.754i)T^{2} \)
73 \( 1 + (1.44 + 1.29i)T + (0.112 + 0.993i)T^{2} \)
79 \( 1 + (-0.212 - 0.977i)T^{2} \)
83 \( 1 + (-1.86 - 0.685i)T + (0.762 + 0.647i)T^{2} \)
89 \( 1 + (-1.97 + 0.300i)T + (0.954 - 0.297i)T^{2} \)
97 \( 1 + (0.0241 - 0.00685i)T + (0.850 - 0.525i)T^{2} \)
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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

−9.389336225243204683413679979665, −8.841456532719304965780187353658, −7.81714413200322735970999438139, −7.37457822487037081456517421166, −6.31813478531198101937497318993, −5.19229263156321332550799719101, −4.12835730608551530021954533408, −3.42484737581149860389489725291, −2.42137725500275158323442819970, −1.30841205380806111060492328323, 0.853626729977989362213351301689, 2.36237294080948937962577058333, 3.40119456319046882406633257449, 4.78506461953347302765451679917, 5.59586943075807066990152625979, 6.10545510222712912076802295118, 7.20402545613683740192844932852, 7.905169996121935787117351305490, 8.431080820806037477243667813406, 9.062116714335698761997584494678

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