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Properties

Label 2-465-15.2-c1-0-3
Degree $2$
Conductor $465$
Sign $-0.994 - 0.105i$
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 + (−1.62 − 0.591i)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 + (1.15 − 3.18i)12-s + (−2.75 + 2.75i)13-s − 0.116·14-s + (2.37 − 3.06i)15-s − 3.73·16-s + (−4.13 + 4.13i)17-s + ⋯
L(s)  = 1  + (0.106 − 0.106i)2-s + (−0.939 − 0.341i)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 + (0.333 − 0.918i)12-s + (−0.762 + 0.762i)13-s − 0.0311·14-s + (0.612 − 0.790i)15-s − 0.932·16-s + (−1.00 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.994 - 0.105i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.994 - 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0160457 + 0.302506i\)
\(L(\frac12)\) \(\approx\) \(0.0160457 + 0.302506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 + 0.591i)T \)
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

−11.41678348735290499801664861951, −10.95075272201497285882699769634, −9.980228276845605720241240860009, −8.418626287031545980454344277684, −7.86506949268247160540521609262, −6.49446021518561686896485752967, −6.38620907641343998685075520283, −4.60447764825668824052905846542, −3.64311890983519045444412106469, −2.35218368520588568320480581373, 0.19191497374825617107588616976, 1.89607583930385578447662192031, 4.25957093982963373781013851122, 4.93111180706127575405374751453, 5.57739300770713288675891005898, 6.80515406982204540462488034800, 7.61542283600486064819207160834, 9.345398488252999170143078060035, 9.632758726247652254798080900850, 10.51283952073496917512507284085

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