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Properties

Label 2-465-15.2-c1-0-11
Degree $2$
Conductor $465$
Sign $-0.674 - 0.738i$
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  + (−1.79 + 1.79i)2-s + (−0.946 + 1.45i)3-s − 4.42i·4-s + (−1.54 − 1.61i)5-s + (−0.903 − 4.29i)6-s + (1.64 + 1.64i)7-s + (4.34 + 4.34i)8-s + (−1.20 − 2.74i)9-s + (5.66 + 0.119i)10-s + 0.702i·11-s + (6.41 + 4.18i)12-s + (3.87 − 3.87i)13-s − 5.91·14-s + (3.80 − 0.717i)15-s − 6.71·16-s + (−0.621 + 0.621i)17-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)2-s + (−0.546 + 0.837i)3-s − 2.21i·4-s + (−0.691 − 0.721i)5-s + (−0.369 − 1.75i)6-s + (0.623 + 0.623i)7-s + (1.53 + 1.53i)8-s + (−0.403 − 0.915i)9-s + (1.79 + 0.0379i)10-s + 0.211i·11-s + (1.85 + 1.20i)12-s + (1.07 − 1.07i)13-s − 1.58·14-s + (0.982 − 0.185i)15-s − 1.67·16-s + (−0.150 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.674 - 0.738i$
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: \((2,\ 465,\ (\ :1/2),\ -0.674 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.213454 + 0.483845i\)
\(L(\frac12)\) \(\approx\) \(0.213454 + 0.483845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.946 - 1.45i)T \)
5 \( 1 + (1.54 + 1.61i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.79 - 1.79i)T - 2iT^{2} \)
7 \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \)
11 \( 1 - 0.702iT - 11T^{2} \)
13 \( 1 + (-3.87 + 3.87i)T - 13iT^{2} \)
17 \( 1 + (0.621 - 0.621i)T - 17iT^{2} \)
19 \( 1 - 3.66iT - 19T^{2} \)
23 \( 1 + (-4.17 - 4.17i)T + 23iT^{2} \)
29 \( 1 - 1.06T + 29T^{2} \)
37 \( 1 + (1.37 + 1.37i)T + 37iT^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (6.40 - 6.40i)T - 43iT^{2} \)
47 \( 1 + (2.39 - 2.39i)T - 47iT^{2} \)
53 \( 1 + (-7.84 - 7.84i)T + 53iT^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + (-7.06 - 7.06i)T + 67iT^{2} \)
71 \( 1 + 0.732iT - 71T^{2} \)
73 \( 1 + (8.32 - 8.32i)T - 73iT^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \)
89 \( 1 - 1.63T + 89T^{2} \)
97 \( 1 + (1.46 + 1.46i)T + 97iT^{2} \)
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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

−11.05267835599741126500999226591, −10.26599613052334023590873296951, −9.290264630959887163521588788689, −8.546371768305693205550040145943, −8.077469828933459807150554930868, −6.89128221221189421519214353701, −5.53910203974065025398552703541, −5.35224441896303895441061657061, −3.82876098042237490677537187518, −1.06011116586908675352727918164, 0.71033338948124235784966249260, 2.00804928799780014280039823382, 3.29960456328931276885992249396, 4.57467507171496896196899836537, 6.60917073088457498767876574204, 7.20132649158925240488810954822, 8.237192240786950010945321337391, 8.763060893447116745036738247869, 10.19151831955329976787097364369, 10.95969087603307876426471192608

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