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Properties

Label 2-465-15.8-c1-0-4
Degree $2$
Conductor $465$
Sign $0.800 + 0.598i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.50 − 1.50i)2-s + (−1.48 − 0.884i)3-s + 2.51i·4-s + (−0.967 − 2.01i)5-s + (0.908 + 3.56i)6-s + (−2.29 + 2.29i)7-s + (0.773 − 0.773i)8-s + (1.43 + 2.63i)9-s + (−1.57 + 4.48i)10-s − 1.64i·11-s + (2.22 − 3.74i)12-s + (0.884 + 0.884i)13-s + 6.90·14-s + (−0.341 + 3.85i)15-s + 2.70·16-s + (−0.210 − 0.210i)17-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)2-s + (−0.859 − 0.510i)3-s + 1.25i·4-s + (−0.432 − 0.901i)5-s + (0.370 + 1.45i)6-s + (−0.869 + 0.869i)7-s + (0.273 − 0.273i)8-s + (0.478 + 0.878i)9-s + (−0.497 + 1.41i)10-s − 0.496i·11-s + (0.642 − 1.08i)12-s + (0.245 + 0.245i)13-s + 1.84·14-s + (−0.0881 + 0.996i)15-s + 0.676·16-s + (−0.0511 − 0.0511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.800 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323409 - 0.107508i\)
\(L(\frac12)\) \(\approx\) \(0.323409 - 0.107508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.48 + 0.884i)T \)
5 \( 1 + (0.967 + 2.01i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.50 + 1.50i)T + 2iT^{2} \)
7 \( 1 + (2.29 - 2.29i)T - 7iT^{2} \)
11 \( 1 + 1.64iT - 11T^{2} \)
13 \( 1 + (-0.884 - 0.884i)T + 13iT^{2} \)
17 \( 1 + (0.210 + 0.210i)T + 17iT^{2} \)
19 \( 1 - 2.54iT - 19T^{2} \)
23 \( 1 + (2.86 - 2.86i)T - 23iT^{2} \)
29 \( 1 - 2.52T + 29T^{2} \)
37 \( 1 + (-6.52 + 6.52i)T - 37iT^{2} \)
41 \( 1 + 1.27iT - 41T^{2} \)
43 \( 1 + (4.59 + 4.59i)T + 43iT^{2} \)
47 \( 1 + (-8.06 - 8.06i)T + 47iT^{2} \)
53 \( 1 + (-1.18 + 1.18i)T - 53iT^{2} \)
59 \( 1 + 6.21T + 59T^{2} \)
61 \( 1 - 3.86T + 61T^{2} \)
67 \( 1 + (-6.83 + 6.83i)T - 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + (-11.4 - 11.4i)T + 73iT^{2} \)
79 \( 1 + 3.25iT - 79T^{2} \)
83 \( 1 + (9.54 - 9.54i)T - 83iT^{2} \)
89 \( 1 - 7.12T + 89T^{2} \)
97 \( 1 + (5.73 - 5.73i)T - 97iT^{2} \)
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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

−11.11562590508683909197760771406, −10.04146210899143923927464875224, −9.277950694478580653625634696266, −8.474600073475029643101545646146, −7.64313486078484292650999823349, −6.18328917743902920606680653374, −5.42134186517857180091440394211, −3.82023618325552700589148642484, −2.31250967699770158593252356259, −0.941391465406310346089383404303, 0.45200801691667643196844461864, 3.32480275975299954230398650953, 4.48474346630374308132150151599, 6.03788628079537820322424332613, 6.66382003889608970384829996904, 7.24295819384693415839018147542, 8.275060972226430386211562734298, 9.539002025785061968409803330390, 10.13410815435766936510059850855, 10.67819224920148727215006353855

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