Properties

Label 2-465-465.119-c1-0-34
Degree $2$
Conductor $465$
Sign $-0.935 + 0.352i$
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  − 1.11·2-s + (−1.73 − 0.0341i)3-s − 0.758·4-s + (1.65 − 1.50i)5-s + (1.92 + 0.0379i)6-s + (−1.74 − 1.00i)7-s + 3.07·8-s + (2.99 + 0.118i)9-s + (−1.83 + 1.67i)10-s + (−0.720 − 1.24i)11-s + (1.31 + 0.0258i)12-s + (0.994 + 1.72i)13-s + (1.94 + 1.12i)14-s + (−2.91 + 2.55i)15-s − 1.90·16-s + (0.447 + 0.258i)17-s + ⋯
L(s)  = 1  − 0.787·2-s + (−0.999 − 0.0196i)3-s − 0.379·4-s + (0.738 − 0.674i)5-s + (0.787 + 0.0155i)6-s + (−0.661 − 0.381i)7-s + 1.08·8-s + (0.999 + 0.0393i)9-s + (−0.581 + 0.531i)10-s + (−0.217 − 0.376i)11-s + (0.379 + 0.00747i)12-s + (0.275 + 0.477i)13-s + (0.520 + 0.300i)14-s + (−0.751 + 0.659i)15-s − 0.476·16-s + (0.108 + 0.0626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0467368 - 0.256704i\)
\(L(\frac12)\) \(\approx\) \(0.0467368 - 0.256704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0341i)T \)
5 \( 1 + (-1.65 + 1.50i)T \)
31 \( 1 + (2.21 + 5.10i)T \)
good2 \( 1 + 1.11T + 2T^{2} \)
7 \( 1 + (1.74 + 1.00i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.720 + 1.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.994 - 1.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.447 - 0.258i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.42 - 2.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.25iT - 23T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
37 \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.50 - 3.17i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.20 + 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.36T + 47T^{2} \)
53 \( 1 + (-7.68 + 4.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.0 + 5.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.17iT - 61T^{2} \)
67 \( 1 + (2.91 - 1.68i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.19 - 3.57i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.44 - 4.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.68 - 2.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 6.15i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.64T + 89T^{2} \)
97 \( 1 + 9.33iT - 97T^{2} \)
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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

−10.31609434721072418389661529531, −9.913183440859836534458307113847, −8.987908726579726024040320350960, −8.109251848597684477503078919067, −6.86002499874449122054184416143, −5.98616562283964622873953292300, −4.96992986982760638280697127273, −3.98529387676704434590024006215, −1.65752293820108796704050135686, −0.24713868549852827185520662925, 1.70522191745222574686279594818, 3.51911382689311167405805548028, 5.06672295307838520227063630309, 5.81354949869479826189705234091, 6.90063023116814084922268483425, 7.66267031974599073310067890970, 9.189256564924179393382218519666, 9.590286968105923617060234221697, 10.54773428744612322422282324067, 10.98631012281809892245915820127

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