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Properties

Label 2-1512-1512.1357-c0-0-1
Degree $2$
Conductor $1512$
Sign $0.448 + 0.893i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
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  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.0603 − 0.342i)5-s + (−0.173 + 0.984i)6-s + (0.766 + 0.642i)7-s + (−0.500 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (0.173 + 0.300i)10-s + (−0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (−0.326 − 0.118i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)18-s + (0.766 − 1.32i)19-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.0603 − 0.342i)5-s + (−0.173 + 0.984i)6-s + (0.766 + 0.642i)7-s + (−0.500 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (0.173 + 0.300i)10-s + (−0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (−0.326 − 0.118i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)18-s + (0.766 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8761653710\)
\(L(\frac12)\) \(\approx\) \(0.8761653710\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.939 + 0.342i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−9.182754842079200878354335947418, −8.701766547343141521173008452868, −7.983472430782914252942031286203, −7.29977213026675343929001813619, −6.62951628354204576525343710642, −5.52479131849199488348580574758, −4.83297147576109621112953991382, −2.93048857429555376776430433512, −2.21556149684753133361914582826, −0.966028394652745167629838444851, 1.60402372327110528461796493493, 2.77046428524558021132973893208, 3.66585119906498387196541395439, 4.57325274169969847879572092652, 5.64263397782294560973694644746, 7.05615873617637285277927350563, 7.57215632724775239327661320581, 8.329215264840815925599672518200, 9.081948671367050772080517774448, 9.983092799013079544120223523519

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