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Properties

Label 2-465-465.374-c0-0-1
Degree $2$
Conductor $465$
Sign $0.800 + 0.599i$
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  + (0.5 + 0.363i)2-s + (0.809 − 0.587i)3-s + (−0.190 − 0.587i)4-s − 5-s + 0.618·6-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.5 − 0.363i)10-s + (−0.5 − 0.363i)12-s + (−0.809 + 0.587i)15-s + (−0.618 + 1.90i)17-s + (0.5 − 0.363i)18-s + (1.30 + 0.951i)19-s + (0.190 + 0.587i)20-s + (−0.190 + 0.587i)23-s + (−0.309 − 0.951i)24-s + ⋯
L(s)  = 1  + (0.5 + 0.363i)2-s + (0.809 − 0.587i)3-s + (−0.190 − 0.587i)4-s − 5-s + 0.618·6-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.5 − 0.363i)10-s + (−0.5 − 0.363i)12-s + (−0.809 + 0.587i)15-s + (−0.618 + 1.90i)17-s + (0.5 − 0.363i)18-s + (1.30 + 0.951i)19-s + (0.190 + 0.587i)20-s + (−0.190 + 0.587i)23-s + (−0.309 − 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.162317985\)
\(L(\frac12)\) \(\approx\) \(1.162317985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−11.25884236383858525075349644964, −10.19797320631320391165254283173, −9.257269887661165692418585675668, −8.248196661559228129908080340334, −7.53037687664239779805333705603, −6.57415191952588966007488319832, −5.60320453543794570246483467647, −4.15695407487105695507077282479, −3.50420858773501775128994078260, −1.57845791781518073905096450400, 2.68029501482923561067883671494, 3.34979100782801195602045840862, 4.49943002105415351046842674801, 5.02696050875600660725499687046, 7.20640943406272869220833045740, 7.66114530167905324722248674862, 8.814545068438454116274185843303, 9.281622731424490167128405915621, 10.68757035081107374563131315338, 11.46307760413485015592787387415

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