Properties

Label 2-465-465.254-c1-0-5
Degree $2$
Conductor $465$
Sign $-0.396 - 0.918i$
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  − 1.73·2-s + (1.58 − 0.689i)3-s + 1.02·4-s + (−1.76 − 1.36i)5-s + (−2.76 + 1.19i)6-s + (−3.83 + 2.21i)7-s + 1.69·8-s + (2.04 − 2.19i)9-s + (3.07 + 2.37i)10-s + (1.12 − 1.94i)11-s + (1.62 − 0.705i)12-s + (−1.39 + 2.41i)13-s + (6.67 − 3.85i)14-s + (−3.75 − 0.953i)15-s − 4.99·16-s + (−4.36 + 2.52i)17-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.917 − 0.398i)3-s + 0.511·4-s + (−0.790 − 0.611i)5-s + (−1.12 + 0.489i)6-s + (−1.45 + 0.837i)7-s + 0.600·8-s + (0.682 − 0.730i)9-s + (0.972 + 0.752i)10-s + (0.338 − 0.586i)11-s + (0.468 − 0.203i)12-s + (−0.386 + 0.668i)13-s + (1.78 − 1.02i)14-s + (−0.969 − 0.246i)15-s − 1.24·16-s + (−1.05 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146555 + 0.222953i\)
\(L(\frac12)\) \(\approx\) \(0.146555 + 0.222953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 + 0.689i)T \)
5 \( 1 + (1.76 + 1.36i)T \)
31 \( 1 + (5.53 + 0.622i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
7 \( 1 + (3.83 - 2.21i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.12 + 1.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.39 - 2.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.36 - 2.52i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.25 - 3.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.41iT - 23T^{2} \)
29 \( 1 + 3.29T + 29T^{2} \)
37 \( 1 + (-3.79 - 6.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.237 - 0.137i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 4.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 + (0.556 + 0.321i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.97 - 5.75i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 + (11.4 + 6.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.93 - 4.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.65 + 4.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.53 + 2.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.1 + 7.01i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + 17.5iT - 97T^{2} \)
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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

−11.27791989723059978011566881118, −9.886558306680384986227102616404, −9.227891127082148426976176705536, −8.844988584923719903583916954960, −7.961153338596154437716912274814, −7.11993653921095795199056972841, −6.07996505946827714474778429820, −4.25430619096103362591725270484, −3.18695199864449895138633348338, −1.62220164538999921604510461397, 0.21962976782365822441814023236, 2.53034685916463731098410527355, 3.65553502731582214463736822527, 4.59969311307353159854392393734, 6.88379660273218268302174464612, 7.20014464663543167460112983074, 8.080247471416823655549636608878, 9.333705659176392932353627278511, 9.503705643465423805855641732070, 10.57278528113769710044495858594

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