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Properties

Label 2-105-5.4-c3-0-7
Degree $2$
Conductor $105$
Sign $0.952 - 0.305i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0765i·2-s + 3i·3-s + 7.99·4-s + (10.6 − 3.42i)5-s − 0.229·6-s − 7i·7-s + 1.22i·8-s − 9·9-s + (0.261 + 0.814i)10-s + 10.8·11-s + 23.9i·12-s − 26.5i·13-s + 0.535·14-s + (10.2 + 31.9i)15-s + 63.8·16-s + 95.1i·17-s + ⋯
L(s)  = 1  + 0.0270i·2-s + 0.577i·3-s + 0.999·4-s + (0.952 − 0.305i)5-s − 0.0156·6-s − 0.377i·7-s + 0.0540i·8-s − 0.333·9-s + (0.00827 + 0.0257i)10-s + 0.298·11-s + 0.576i·12-s − 0.566i·13-s + 0.0102·14-s + (0.176 + 0.549i)15-s + 0.997·16-s + 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.952 - 0.305i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.18712 + 0.342787i\)
\(L(\frac12)\) \(\approx\) \(2.18712 + 0.342787i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 + (-10.6 + 3.42i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 0.0765iT - 8T^{2} \)
11 \( 1 - 10.8T + 1.33e3T^{2} \)
13 \( 1 + 26.5iT - 2.19e3T^{2} \)
17 \( 1 - 95.1iT - 4.91e3T^{2} \)
19 \( 1 - 35.4T + 6.85e3T^{2} \)
23 \( 1 - 62.8iT - 1.21e4T^{2} \)
29 \( 1 + 117.T + 2.43e4T^{2} \)
31 \( 1 + 171.T + 2.97e4T^{2} \)
37 \( 1 + 203. iT - 5.06e4T^{2} \)
41 \( 1 + 428.T + 6.89e4T^{2} \)
43 \( 1 + 96.5iT - 7.95e4T^{2} \)
47 \( 1 - 407. iT - 1.03e5T^{2} \)
53 \( 1 + 380. iT - 1.48e5T^{2} \)
59 \( 1 - 287.T + 2.05e5T^{2} \)
61 \( 1 + 823.T + 2.26e5T^{2} \)
67 \( 1 + 585. iT - 3.00e5T^{2} \)
71 \( 1 + 653.T + 3.57e5T^{2} \)
73 \( 1 + 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 - 751.T + 4.93e5T^{2} \)
83 \( 1 - 844. iT - 5.71e5T^{2} \)
89 \( 1 - 262.T + 7.04e5T^{2} \)
97 \( 1 - 814. iT - 9.12e5T^{2} \)
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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

−13.32026075654490803870792892766, −12.28313193006960994558381139873, −10.96398792364975980526036805822, −10.28608095673956434256690666908, −9.182110673440796403488486467184, −7.76558357414562637450680132924, −6.34932276592055039148174377597, −5.37152623328351473047664814764, −3.50984269076263304066346021494, −1.73747017558678189148958473993, 1.70016659295538884963064976276, 2.91051604422010225634448494078, 5.41532027860132542952018560008, 6.54831975672710625107499146607, 7.31743099554084264232677294599, 8.941489090649254838130610374462, 10.07404859908963552608168067534, 11.32618516647186834349583593988, 12.03657092853511566990620403297, 13.26452979087272330708506488069

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