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Properties

Label 2-465-155.4-c1-0-6
Degree $2$
Conductor $465$
Sign $0.471 - 0.882i$
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  + (−0.932 + 0.302i)2-s + (−0.951 − 0.309i)3-s + (−0.840 + 0.610i)4-s + (0.949 + 2.02i)5-s + 0.980·6-s + (−2.34 − 3.22i)7-s + (1.75 − 2.41i)8-s + (0.809 + 0.587i)9-s + (−1.49 − 1.59i)10-s + (2.64 − 1.91i)11-s + (0.987 − 0.321i)12-s + (2.34 + 0.761i)13-s + (3.16 + 2.29i)14-s + (−0.277 − 2.21i)15-s + (−0.260 + 0.802i)16-s + (−2.03 + 2.79i)17-s + ⋯
L(s)  = 1  + (−0.659 + 0.214i)2-s + (−0.549 − 0.178i)3-s + (−0.420 + 0.305i)4-s + (0.424 + 0.905i)5-s + 0.400·6-s + (−0.885 − 1.21i)7-s + (0.619 − 0.852i)8-s + (0.269 + 0.195i)9-s + (−0.473 − 0.505i)10-s + (0.796 − 0.578i)11-s + (0.285 − 0.0926i)12-s + (0.650 + 0.211i)13-s + (0.844 + 0.613i)14-s + (−0.0717 − 0.572i)15-s + (−0.0651 + 0.200i)16-s + (−0.492 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.471 - 0.882i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.471 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602601 + 0.361304i\)
\(L(\frac12)\) \(\approx\) \(0.602601 + 0.361304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.949 - 2.02i)T \)
31 \( 1 + (-3.49 - 4.33i)T \)
good2 \( 1 + (0.932 - 0.302i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (2.34 + 3.22i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-2.64 + 1.91i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.34 - 0.761i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.03 - 2.79i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.35 - 7.24i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.477 + 0.656i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.71 + 5.26i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 11.5iT - 37T^{2} \)
41 \( 1 + (-0.833 - 2.56i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-5.68 + 1.84i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-11.5 - 3.73i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.59 + 4.94i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.429 + 1.32i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 1.54T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 + (-5.31 - 3.86i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.16 - 7.10i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (10.5 + 7.69i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.31 - 0.753i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.897 + 0.651i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-8.37 - 11.5i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.88221948753884610405184664790, −10.19367154925471509344912610388, −9.636257360254492550459178384847, −8.437198706570550606240680536573, −7.45517040956390496701298596096, −6.59205460011147550532421754579, −6.04746818357376831866576325292, −4.12609610003295415997341768316, −3.45827557997414289870044073190, −1.15965417975107166244625331634, 0.73761570835284916259107379831, 2.34443927027838455912259737156, 4.27588137098580830112867614896, 5.30871197196183912447329155534, 5.94933274184397151969632729511, 7.20109206688477551646298069893, 8.893233666221677412352308205100, 9.078860809836327289590359094054, 9.647800799166242850797560295038, 10.80042419439051991733452701484

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