Properties

Degree $2$
Conductor $6012$
Sign $0.0713 - 0.997i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.47·5-s + 3.93·7-s − 4.34i·11-s + 0.788i·13-s − 5.32·17-s + 4.26·19-s − 6.69·23-s + 7.06·25-s + 5.99i·29-s − 6.79·31-s − 13.6·35-s + 4.66i·37-s + 4.62·41-s − 11.6i·43-s + 12.6i·47-s + ⋯
L(s)  = 1  − 1.55·5-s + 1.48·7-s − 1.30i·11-s + 0.218i·13-s − 1.29·17-s + 0.977·19-s − 1.39·23-s + 1.41·25-s + 1.11i·29-s − 1.22·31-s − 2.30·35-s + 0.766i·37-s + 0.721·41-s − 1.76i·43-s + 1.84i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $0.0713 - 0.997i$
Motivic weight: \(1\)
Character: $\chi_{6012} (3005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 0.0713 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8776382258\)
\(L(\frac12)\) \(\approx\) \(0.8776382258\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 + (-9.99 - 8.19i)T \)
good5 \( 1 + 3.47T + 5T^{2} \)
7 \( 1 - 3.93T + 7T^{2} \)
11 \( 1 + 4.34iT - 11T^{2} \)
13 \( 1 - 0.788iT - 13T^{2} \)
17 \( 1 + 5.32T + 17T^{2} \)
19 \( 1 - 4.26T + 19T^{2} \)
23 \( 1 + 6.69T + 23T^{2} \)
29 \( 1 - 5.99iT - 29T^{2} \)
31 \( 1 + 6.79T + 31T^{2} \)
37 \( 1 - 4.66iT - 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 - 0.406T + 53T^{2} \)
59 \( 1 - 6.56T + 59T^{2} \)
61 \( 1 + 4.94T + 61T^{2} \)
67 \( 1 - 3.40iT - 67T^{2} \)
71 \( 1 + 0.507T + 71T^{2} \)
73 \( 1 + 2.74iT - 73T^{2} \)
79 \( 1 - 7.44iT - 79T^{2} \)
83 \( 1 - 6.51T + 83T^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + 3.32T + 97T^{2} \)
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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

−8.318012249256576564921973735636, −7.57356277016944138790082762048, −7.14713194232946056648879598846, −6.06393776310424086563825716030, −5.26279769383648455697363029168, −4.51866316561165461932071651179, −3.91322787071430773813505698391, −3.19733145054989238613163082253, −2.01172930308173900793595229433, −0.917277214608108519832613330847, 0.27323167007814899422168834942, 1.67427098345048673315555730682, 2.42574315204991147454055506760, 3.77171684952342518173328485083, 4.26944119736629776272123161551, 4.79017943400319099858165200650, 5.60141274397612528051060621432, 6.74926837850963632648289741906, 7.49024545478336800927724523816, 7.80763241620207704485660048584

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