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Properties

Label 2-945-945.349-c0-0-1
Degree $2$
Conductor $945$
Sign $-0.230 + 0.973i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
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  + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (0.939 − 0.342i)7-s + (−0.499 − 0.866i)9-s + (0.266 + 0.223i)11-s + (−0.766 + 0.642i)12-s + (−0.266 − 1.50i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 − 1.62i)17-s + (0.939 − 0.342i)20-s + (0.173 − 0.984i)21-s + (0.173 − 0.984i)25-s − 0.999·27-s − 28-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (0.939 − 0.342i)7-s + (−0.499 − 0.866i)9-s + (0.266 + 0.223i)11-s + (−0.766 + 0.642i)12-s + (−0.266 − 1.50i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 − 1.62i)17-s + (0.939 − 0.342i)20-s + (0.173 − 0.984i)21-s + (0.173 − 0.984i)25-s − 0.999·27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ -0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8269316783\)
\(L(\frac12)\) \(\approx\) \(0.8269316783\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
good2 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.993899987831188374377567494705, −8.931619122255027112543018558226, −8.308582252415898042789545393023, −7.48025251201126019102740886009, −6.97244569822028270531510974265, −5.59603494239150041609488320443, −4.67722908981144224361110799877, −3.62179227077036901613294085216, −2.51877746362251426820520669849, −0.817014751694380088392470896181, 1.97024267185447380444539515583, 3.66185918125260496595052208925, 4.36997013782555147661907860433, 4.72147938547101985773567340101, 5.95481556747389398576533842984, 7.56895006480881413136784744081, 8.246076819254342154728077814956, 8.953636557152935330480208326162, 9.184843629777538629433587885296, 10.43837046159618300067656532773

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