Properties

Label 2-6012-501.500-c1-0-55
Degree $2$
Conductor $6012$
Sign $0.606 - 0.795i$
Analytic cond. $48.0060$
Root an. cond. $6.92864$
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  − 3.84·5-s − 2.14·7-s − 0.389i·11-s − 5.51i·13-s − 6.90·17-s − 6.97·19-s − 4.97·23-s + 9.78·25-s − 7.02i·29-s + 0.566·31-s + 8.25·35-s + 7.77i·37-s − 8.87·41-s − 3.56i·43-s − 12.8i·47-s + ⋯
L(s)  = 1  − 1.71·5-s − 0.811·7-s − 0.117i·11-s − 1.52i·13-s − 1.67·17-s − 1.59·19-s − 1.03·23-s + 1.95·25-s − 1.30i·29-s + 0.101·31-s + 1.39·35-s + 1.27i·37-s − 1.38·41-s − 0.544i·43-s − 1.86i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(48.0060\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6012} (3005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02145335527\)
\(L(\frac12)\) \(\approx\) \(0.02145335527\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.86 - 12.3i)T \)
good5 \( 1 + 3.84T + 5T^{2} \)
7 \( 1 + 2.14T + 7T^{2} \)
11 \( 1 + 0.389iT - 11T^{2} \)
13 \( 1 + 5.51iT - 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + 7.02iT - 29T^{2} \)
31 \( 1 - 0.566T + 31T^{2} \)
37 \( 1 - 7.77iT - 37T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 + 3.56iT - 43T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 - 2.28T + 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 + 2.35T + 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 - 6.60iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−7.64501463281675814847578886691, −6.53723323132869053360360790031, −6.46790126997157454261236413091, −5.17697243207856816410528552211, −4.37052861237135553356290689748, −3.78452454805353681750364441766, −3.10847712423457821365313942874, −2.14591292312197475680469181381, −0.30154698691227104481253567227, −0.01744623214716556780844733827, 1.74360845544807508308421926573, 2.74393107494041061438243185610, 3.76435200391774732618738472303, 4.26561085697560262456035970816, 4.68556882140697306114950093482, 6.11539905483958893328992775590, 6.75396453427274119558809446455, 7.08833948171089315666325457996, 8.022568773929599490504407592075, 8.727260424623320337430878792581

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