Properties

Label 2-6012-1.1-c1-0-24
Degree $2$
Conductor $6012$
Sign $-1$
Analytic cond. $48.0060$
Root an. cond. $6.92864$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.21·5-s − 5.13·7-s + 1.59·11-s + 5.05·13-s − 0.754·17-s − 1.66·19-s − 6.04·23-s + 12.7·25-s + 4.24·29-s − 1.07·31-s + 21.6·35-s + 4.94·37-s − 3.04·41-s + 12.5·43-s − 9.35·47-s + 19.3·49-s + 8.52·53-s − 6.70·55-s + 3.98·59-s + 8.18·61-s − 21.2·65-s − 4.99·67-s − 6.73·71-s − 6.06·73-s − 8.17·77-s − 13.5·79-s + 5.95·83-s + ⋯
L(s)  = 1  − 1.88·5-s − 1.94·7-s + 0.479·11-s + 1.40·13-s − 0.183·17-s − 0.382·19-s − 1.25·23-s + 2.54·25-s + 0.788·29-s − 0.192·31-s + 3.65·35-s + 0.813·37-s − 0.475·41-s + 1.91·43-s − 1.36·47-s + 2.77·49-s + 1.17·53-s − 0.903·55-s + 0.518·59-s + 1.04·61-s − 2.64·65-s − 0.610·67-s − 0.798·71-s − 0.709·73-s − 0.931·77-s − 1.52·79-s + 0.653·83-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 + 4.21T + 5T^{2} \)
7 \( 1 + 5.13T + 7T^{2} \)
11 \( 1 - 1.59T + 11T^{2} \)
13 \( 1 - 5.05T + 13T^{2} \)
17 \( 1 + 0.754T + 17T^{2} \)
19 \( 1 + 1.66T + 19T^{2} \)
23 \( 1 + 6.04T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 - 4.94T + 37T^{2} \)
41 \( 1 + 3.04T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 - 8.52T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 - 8.18T + 61T^{2} \)
67 \( 1 + 4.99T + 67T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 + 6.06T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 5.95T + 83T^{2} \)
89 \( 1 + 7.83T + 89T^{2} \)
97 \( 1 - 3.42T + 97T^{2} \)
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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

−7.65959143771694010954566514246, −7.02423144739531811146872466489, −6.34980000135258019231433358303, −5.88593285336930502978066217820, −4.43841978127146667358795954354, −3.87885962340186311344315105050, −3.48903622527512118185334490043, −2.65013601507624858897581298123, −0.931786599871272815703546304316, 0, 0.931786599871272815703546304316, 2.65013601507624858897581298123, 3.48903622527512118185334490043, 3.87885962340186311344315105050, 4.43841978127146667358795954354, 5.88593285336930502978066217820, 6.34980000135258019231433358303, 7.02423144739531811146872466489, 7.65959143771694010954566514246

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