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Properties

Label 2-3069-341.278-c0-0-0
Degree $2$
Conductor $3069$
Sign $-0.794 - 0.606i$
Analytic cond. $1.53163$
Root an. cond. $1.23759$
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  + (−1.30 + 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + 1.61·10-s + (−0.951 − 0.309i)11-s + (−0.587 − 0.809i)13-s + (−0.499 − 1.53i)14-s + (0.951 − 1.30i)17-s + (−1.30 + 0.951i)20-s + (1.53 − 0.499i)22-s + 1.61i·23-s + (1.53 + 0.5i)26-s + (1.30 + 0.951i)28-s + (−0.587 − 0.190i)29-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + 1.61·10-s + (−0.951 − 0.309i)11-s + (−0.587 − 0.809i)13-s + (−0.499 − 1.53i)14-s + (0.951 − 1.30i)17-s + (−1.30 + 0.951i)20-s + (1.53 − 0.499i)22-s + 1.61i·23-s + (1.53 + 0.5i)26-s + (1.30 + 0.951i)28-s + (−0.587 − 0.190i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3069\)    =    \(3^{2} \cdot 11 \cdot 31\)
Sign: $-0.794 - 0.606i$
Analytic conductor: \(1.53163\)
Root analytic conductor: \(1.23759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3069} (2665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3069,\ (\ :0),\ -0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2535542133\)
\(L(\frac12)\) \(\approx\) \(0.2535542133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
good2 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.61iT - T^{2} \)
29 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.119272227960825363721981390616, −8.291197925475405828141452697941, −7.69453705015340843358470588557, −7.45120052596838557540029526650, −6.29150795954412143297019945982, −5.37444358142129514049070959033, −5.13459683596078091549899443059, −3.52390698900113861079359532691, −2.61341267656107250828460673687, −0.993886732549319956964329871475, 0.29867857044336440412353208377, 1.77767250908650464485496770269, 2.71215713454088288102479133771, 3.62299006049498412562529434817, 4.27224228421158183220064988086, 5.55597845420818159164000756557, 6.91881896523529384017115004287, 7.22431981359189494174172147282, 8.106437268297738964955598291871, 8.468056003091619911953639753117

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