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Properties

Label 2-465-155.4-c1-0-2
Degree $2$
Conductor $465$
Sign $-0.793 - 0.608i$
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  + (0.464 − 0.150i)2-s + (0.951 + 0.309i)3-s + (−1.42 + 1.03i)4-s + (−2.22 − 0.252i)5-s + 0.487·6-s + (0.802 + 1.10i)7-s + (−1.07 + 1.48i)8-s + (0.809 + 0.587i)9-s + (−1.06 + 0.218i)10-s + (−3.23 + 2.34i)11-s + (−1.67 + 0.544i)12-s + (−4.08 − 1.32i)13-s + (0.538 + 0.391i)14-s + (−2.03 − 0.926i)15-s + (0.812 − 2.49i)16-s + (−0.591 + 0.814i)17-s + ⋯
L(s)  = 1  + (0.328 − 0.106i)2-s + (0.549 + 0.178i)3-s + (−0.712 + 0.517i)4-s + (−0.993 − 0.112i)5-s + 0.199·6-s + (0.303 + 0.417i)7-s + (−0.381 + 0.525i)8-s + (0.269 + 0.195i)9-s + (−0.338 + 0.0689i)10-s + (−0.973 + 0.707i)11-s + (−0.483 + 0.157i)12-s + (−1.13 − 0.368i)13-s + (0.144 + 0.104i)14-s + (−0.525 − 0.239i)15-s + (0.203 − 0.624i)16-s + (−0.143 + 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.793 - 0.608i$
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: \((2,\ 465,\ (\ :1/2),\ -0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235710 + 0.694893i\)
\(L(\frac12)\) \(\approx\) \(0.235710 + 0.694893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.252i)T \)
31 \( 1 + (-4.87 - 2.68i)T \)
good2 \( 1 + (-0.464 + 0.150i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-0.802 - 1.10i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (3.23 - 2.34i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.08 + 1.32i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.591 - 0.814i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.11 + 3.43i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.27 - 7.25i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.99 - 6.15i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 1.93iT - 37T^{2} \)
41 \( 1 + (0.361 + 1.11i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (9.21 - 2.99i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-10.3 - 3.37i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.874 + 1.20i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.544 + 1.67i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 9.09T + 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + (12.9 + 9.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.94 - 6.80i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.51 - 3.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.52 - 2.12i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.10 - 2.98i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.20 - 5.78i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.69787571240100797705923710134, −10.43136840952362736434518405227, −9.508216281037009788620437766803, −8.525882732164421815647437568954, −7.910459442703918093120774510526, −7.17474846365237278639711333602, −5.18387876992981112471769535031, −4.69369249463086590557198450722, −3.53050095282283799195844874606, −2.50167045676766143761646876090, 0.37339559677098697621312540839, 2.57869872975971757694541431371, 4.02353625098383050069074314858, 4.61261310529383257224416701588, 5.91126107526884828978215068810, 7.11299691723664028875174488736, 8.109069229152899963038820019993, 8.585481729347966725356068082500, 9.978912922790707071904643394320, 10.44931636548439199908898522786

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