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Properties

Label 2-465-465.149-c0-0-1
Degree $2$
Conductor $465$
Sign $0.920 - 0.390i$
Analytic cond. $0.232065$
Root an. cond. $0.481731$
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  + 2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + 0.999·15-s − 16-s + (−0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s − 2·23-s + (−0.5 − 0.866i)24-s + (−0.499 − 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + 0.999·15-s − 16-s + (−0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s − 2·23-s + (−0.5 − 0.866i)24-s + (−0.499 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438546988\)
\(L(\frac12)\) \(\approx\) \(1.438546988\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
good2 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.59852518388581688746245574029, −10.15752450986497744014784728000, −9.586733195601051727789709324745, −8.748181028621235264762097148371, −7.926296929739418307513682917488, −6.17090188738592773646433912542, −5.35434783178590624372577046138, −4.53759058082505254962062469548, −3.73926387856976216355376985572, −2.38649529682368923708268769367, 2.17454843984644904090207735470, 3.17655406414512934515884898783, 4.26366762328766929649588131971, 5.80389257627190025658838562794, 6.32087329780485451951345529758, 7.33142157226543522467694453456, 8.431081848156009762445608574013, 9.351394053328515835374978861946, 10.34995362288077495315283956763, 11.57426140620398920506855411896

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