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Properties

Label 2-465-15.8-c1-0-1
Degree $2$
Conductor $465$
Sign $0.510 - 0.859i$
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 + (−1.12 − 1.31i)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 + (−2.09 + 1.80i)12-s + (−2.01 − 2.01i)13-s − 0.149·14-s + (3.87 − 0.0687i)15-s − 1.73·16-s + (0.865 + 0.865i)17-s + ⋯
L(s)  = 1  + (−0.318 − 0.318i)2-s + (−0.652 − 0.758i)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 + (−0.604 + 0.519i)12-s + (−0.559 − 0.559i)13-s − 0.0400·14-s + (0.999 − 0.0177i)15-s − 0.433·16-s + (0.209 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.510 - 0.859i$
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: \((2,\ 465,\ (\ :1/2),\ 0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335550 + 0.191006i\)
\(L(\frac12)\) \(\approx\) \(0.335550 + 0.191006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 + 1.31i)T \)
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

−11.14551185010754872080993905451, −10.35151912992183631273474588192, −9.840598819857605205033661216944, −8.201734946759053956137651504935, −7.60144897048751511749336713477, −6.46266795456649890988123838551, −5.77682266049448014309467641882, −4.55852945164061491914411582468, −2.84063510695134306005671742690, −1.47281368820940056538321809743, 0.29558915722647777343648546282, 3.04233709568080472278898875734, 4.28768137537508247800797936171, 4.89739240427185548438349114944, 6.32298262304108531434080649326, 7.24247295147484698677123631562, 8.380406054249377778993789521721, 8.970664167833059801990702607743, 9.819382062705300341145504099148, 11.03835667150189027730007099191

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