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Properties

Label 2-465-15.2-c1-0-15
Degree $2$
Conductor $465$
Sign $-0.376 - 0.926i$
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  + (0.450 − 0.450i)2-s + (−1.31 + 1.12i)3-s + 1.59i·4-s + (1.42 + 1.72i)5-s + (−0.0827 + 1.09i)6-s + (0.166 + 0.166i)7-s + (1.61 + 1.61i)8-s + (0.449 − 2.96i)9-s + (1.41 + 0.132i)10-s + 1.30i·11-s + (−1.80 − 2.09i)12-s + (−2.01 + 2.01i)13-s + 0.149·14-s + (−3.81 − 0.647i)15-s − 1.73·16-s + (−0.865 + 0.865i)17-s + ⋯
L(s)  = 1  + (0.318 − 0.318i)2-s + (−0.758 + 0.652i)3-s + 0.797i·4-s + (0.638 + 0.769i)5-s + (−0.0337 + 0.448i)6-s + (0.0629 + 0.0629i)7-s + (0.572 + 0.572i)8-s + (0.149 − 0.988i)9-s + (0.448 + 0.0417i)10-s + 0.394i·11-s + (−0.519 − 0.604i)12-s + (−0.559 + 0.559i)13-s + 0.0400·14-s + (−0.985 − 0.167i)15-s − 0.433·16-s + (−0.209 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.376 - 0.926i$
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: \((2,\ 465,\ (\ :1/2),\ -0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733085 + 1.08868i\)
\(L(\frac12)\) \(\approx\) \(0.733085 + 1.08868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.31 - 1.12i)T \)
5 \( 1 + (-1.42 - 1.72i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.450 + 0.450i)T - 2iT^{2} \)
7 \( 1 + (-0.166 - 0.166i)T + 7iT^{2} \)
11 \( 1 - 1.30iT - 11T^{2} \)
13 \( 1 + (2.01 - 2.01i)T - 13iT^{2} \)
17 \( 1 + (0.865 - 0.865i)T - 17iT^{2} \)
19 \( 1 + 6.62iT - 19T^{2} \)
23 \( 1 + (-3.49 - 3.49i)T + 23iT^{2} \)
29 \( 1 + 8.04T + 29T^{2} \)
37 \( 1 + (-1.16 - 1.16i)T + 37iT^{2} \)
41 \( 1 - 7.29iT - 41T^{2} \)
43 \( 1 + (-1.00 + 1.00i)T - 43iT^{2} \)
47 \( 1 + (-5.02 + 5.02i)T - 47iT^{2} \)
53 \( 1 + (-5.89 - 5.89i)T + 53iT^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 - 3.40T + 61T^{2} \)
67 \( 1 + (6.19 + 6.19i)T + 67iT^{2} \)
71 \( 1 + 7.53iT - 71T^{2} \)
73 \( 1 + (5.11 - 5.11i)T - 73iT^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + (-4.83 - 4.83i)T + 97iT^{2} \)
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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

−11.35537179340514094142327126456, −10.66721549850877642141441593403, −9.615885278193293929001258815124, −8.948919427115179737842827765455, −7.33259966783321149973627068828, −6.76713941478327432946851574908, −5.45050568820944595457686595137, −4.56034634042917593804725971663, −3.45151170912813278992034044771, −2.25953869116397214201119041860, 0.815674252515875664473839047900, 2.08538339876606883647659324906, 4.33518051754718885188795608285, 5.48821619432690334879633802251, 5.72455601540118434170948981959, 6.84045528693669715804033832990, 7.82040503732320432161269763509, 9.002670860866992399771346537192, 10.07802786714549154016410602441, 10.66308340374997849052441439191

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