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Properties

Label 2-465-15.8-c1-0-48
Degree $2$
Conductor $465$
Sign $-0.557 + 0.830i$
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  + (−0.150 − 0.150i)2-s + (0.591 − 1.62i)3-s − 1.95i·4-s + (0.684 + 2.12i)5-s + (−0.333 + 0.155i)6-s + (−0.387 + 0.387i)7-s + (−0.594 + 0.594i)8-s + (−2.30 − 1.92i)9-s + (0.217 − 0.422i)10-s − 5.94i·11-s + (−3.18 − 1.15i)12-s + (−2.75 − 2.75i)13-s + 0.116·14-s + (3.87 + 0.145i)15-s − 3.73·16-s + (4.13 + 4.13i)17-s + ⋯
L(s)  = 1  + (−0.106 − 0.106i)2-s + (0.341 − 0.939i)3-s − 0.977i·4-s + (0.306 + 0.952i)5-s + (−0.136 + 0.0636i)6-s + (−0.146 + 0.146i)7-s + (−0.210 + 0.210i)8-s + (−0.766 − 0.642i)9-s + (0.0686 − 0.133i)10-s − 1.79i·11-s + (−0.918 − 0.333i)12-s + (−0.762 − 0.762i)13-s + 0.0311·14-s + (0.999 + 0.0375i)15-s − 0.932·16-s + (1.00 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.557 + 0.830i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629673 - 1.18099i\)
\(L(\frac12)\) \(\approx\) \(0.629673 - 1.18099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.591 + 1.62i)T \)
5 \( 1 + (-0.684 - 2.12i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.150 + 0.150i)T + 2iT^{2} \)
7 \( 1 + (0.387 - 0.387i)T - 7iT^{2} \)
11 \( 1 + 5.94iT - 11T^{2} \)
13 \( 1 + (2.75 + 2.75i)T + 13iT^{2} \)
17 \( 1 + (-4.13 - 4.13i)T + 17iT^{2} \)
19 \( 1 + 1.21iT - 19T^{2} \)
23 \( 1 + (-6.27 + 6.27i)T - 23iT^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
37 \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \)
41 \( 1 - 6.88iT - 41T^{2} \)
43 \( 1 + (4.32 + 4.32i)T + 43iT^{2} \)
47 \( 1 + (-3.93 - 3.93i)T + 47iT^{2} \)
53 \( 1 + (-4.51 + 4.51i)T - 53iT^{2} \)
59 \( 1 - 4.65T + 59T^{2} \)
61 \( 1 - 0.744T + 61T^{2} \)
67 \( 1 + (-5.51 + 5.51i)T - 67iT^{2} \)
71 \( 1 - 4.91iT - 71T^{2} \)
73 \( 1 + (-9.51 - 9.51i)T + 73iT^{2} \)
79 \( 1 + 3.31iT - 79T^{2} \)
83 \( 1 + (4.12 - 4.12i)T - 83iT^{2} \)
89 \( 1 - 3.87T + 89T^{2} \)
97 \( 1 + (-5.32 + 5.32i)T - 97iT^{2} \)
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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

−10.83186704275749667369293919223, −9.930626409205057125806077868500, −8.899036001887244604179792235418, −8.019951382702062191265160046765, −6.89276251926674929773128463263, −6.06719380208397656109171909710, −5.48242278473872746015097426851, −3.30647120763197370976273634839, −2.45306789476102640941240376740, −0.829354877259097860391857111216, 2.20604981849998535712522935914, 3.59478988474007648165484587460, 4.62792422450025167777585146500, 5.25312522945877134142654137435, 7.11721382658004879601092309529, 7.69265291389697483609648601175, 8.913071074280224945145846362750, 9.516426600481145434774016013902, 9.992538920774312603659585185186, 11.57180210067856902217458354127

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