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Properties

Label 2-755-5.4-c1-0-49
Degree $2$
Conductor $755$
Sign $0.936 + 0.350i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
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  + 1.84i·2-s + 2.98i·3-s − 1.39·4-s + (−2.09 − 0.784i)5-s − 5.49·6-s − 5.05i·7-s + 1.11i·8-s − 5.90·9-s + (1.44 − 3.85i)10-s − 3.23·11-s − 4.15i·12-s − 2.63i·13-s + 9.31·14-s + (2.34 − 6.25i)15-s − 4.84·16-s − 0.962i·17-s + ⋯
L(s)  = 1  + 1.30i·2-s + 1.72i·3-s − 0.696·4-s + (−0.936 − 0.350i)5-s − 2.24·6-s − 1.91i·7-s + 0.395i·8-s − 1.96·9-s + (0.456 − 1.21i)10-s − 0.974·11-s − 1.19i·12-s − 0.731i·13-s + 2.48·14-s + (0.604 − 1.61i)15-s − 1.21·16-s − 0.233i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162792 - 0.0294775i\)
\(L(\frac12)\) \(\approx\) \(0.162792 - 0.0294775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.09 + 0.784i)T \)
151 \( 1 + T \)
good2 \( 1 - 1.84iT - 2T^{2} \)
3 \( 1 - 2.98iT - 3T^{2} \)
7 \( 1 + 5.05iT - 7T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + 2.63iT - 13T^{2} \)
17 \( 1 + 0.962iT - 17T^{2} \)
19 \( 1 - 2.66T + 19T^{2} \)
23 \( 1 - 5.80iT - 23T^{2} \)
29 \( 1 + 2.13T + 29T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + 7.66iT - 37T^{2} \)
41 \( 1 + 9.30T + 41T^{2} \)
43 \( 1 + 5.98iT - 43T^{2} \)
47 \( 1 + 7.77iT - 47T^{2} \)
53 \( 1 - 4.36iT - 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 - 2.79iT - 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + 4.34T + 79T^{2} \)
83 \( 1 + 8.54iT - 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 6.10iT - 97T^{2} \)
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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

−10.30333875360952798781709472817, −9.387615557818995146439873288163, −8.389562760699132963562659164642, −7.58670052591356537836982618873, −7.15820736795649262249584893208, −5.43910795319931736123141901347, −5.07220722375609586393717253738, −4.00345946960510397694997152680, −3.40940501466438605523173346494, −0.080161995680484824332040155426, 1.69944019164978905890006308332, 2.54280845588989417291192684690, 3.19229303221721661790313441936, 4.93967457959929073612613138867, 6.18306263988617094061904019054, 6.91057718280963438513554612564, 7.976709364191828230065297476507, 8.568841900875826006637661958777, 9.537959346595162183099081700923, 10.88502648720345787207292537336

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