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Properties

Label 2-465-31.25-c1-0-2
Degree $2$
Conductor $465$
Sign $-0.425 - 0.905i$
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  + (0.5 + 0.866i)3-s − 2·4-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (−2.21 + 3.83i)11-s + (−1 − 1.73i)12-s + (−2.71 + 4.69i)13-s + 0.999·15-s + 4·16-s + (2 + 3.46i)17-s + (−0.712 − 1.23i)19-s + (−1 + 1.73i)20-s + 2·23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 4-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (−0.667 + 1.15i)11-s + (−0.288 − 0.499i)12-s + (−0.752 + 1.30i)13-s + 0.258·15-s + 16-s + (0.485 + 0.840i)17-s + (−0.163 − 0.283i)19-s + (−0.223 + 0.387i)20-s + 0.417·23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.425 - 0.905i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.425 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480565 + 0.756769i\)
\(L(\frac12)\) \(\approx\) \(0.480565 + 0.756769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.212 - 5.56i)T \)
good2 \( 1 + 2T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.21 - 3.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.71 - 4.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.712 + 1.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
37 \( 1 + (-0.712 - 1.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.21 - 7.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.71 - 2.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + (-4.42 + 7.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.21 + 3.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + (-6.42 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.21 - 2.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.71 + 9.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.63 - 14.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.57T + 89T^{2} \)
97 \( 1 + 2.57T + 97T^{2} \)
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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

−11.25232189500583254048596461163, −9.908740002824075576399953419215, −9.757046747573698550065854373431, −8.749990911207448675009962213177, −7.936569090081237200770361831714, −6.76448896791493598912108967222, −5.19512268977801431460449332679, −4.72381833816772949957870296050, −3.64899686367562349721175876404, −1.95234936830704850753151687512, 0.54341911723181272273844315387, 2.67009180153872785290152770198, 3.63618953659879723517234296610, 5.25459509371244060599408044133, 5.79726004704278767562358579794, 7.35328839442690772655642346173, 7.993501936155727929224223702210, 8.900610819578990680327762085599, 9.837993556082481149610591784981, 10.59684497536894135344354483303

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