Properties

Label 2-2352-588.251-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.995 - 0.0960i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 + 0.433i)3-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.678 − 0.541i)13-s + 1.24·19-s + (0.900 − 0.433i)21-s + (−0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + 0.445·31-s + (−0.0990 + 0.433i)37-s + (−0.376 − 0.781i)39-s + (0.678 + 1.40i)43-s + (−0.222 − 0.974i)49-s + (1.12 + 0.541i)57-s + (−1.90 − 0.433i)61-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)3-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.678 − 0.541i)13-s + 1.24·19-s + (0.900 − 0.433i)21-s + (−0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + 0.445·31-s + (−0.0990 + 0.433i)37-s + (−0.376 − 0.781i)39-s + (0.678 + 1.40i)43-s + (−0.222 − 0.974i)49-s + (1.12 + 0.541i)57-s + (−1.90 − 0.433i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.995 - 0.0960i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.995 - 0.0960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.751829171\)
\(L(\frac12)\) \(\approx\) \(1.751829171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
good5 \( 1 + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.900 - 0.433i)T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - 0.445T + T^{2} \)
37 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (1.90 + 0.433i)T + (0.900 + 0.433i)T^{2} \)
67 \( 1 - 0.867iT - T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.52 - 1.21i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + 1.94iT - T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−9.213382902528685623944695872790, −8.319405044629797066379231447373, −7.66866800146716269980695574243, −7.27677687528566632974000548620, −6.02223946913005617378765071605, −4.92354595125333116942223091019, −4.41905042605301835567597193727, −3.40419593127907856301500176221, −2.58905056271359823930045397405, −1.34581639210441885381041289027, 1.48657834125496618821922384319, 2.35756799934450588974114719000, 3.23809845493727868852194853221, 4.27873992943529624391779349223, 5.22248942342986568734996506745, 6.04966554863206445841782250330, 7.15792042265626909352049226642, 7.59785251265433126890598774089, 8.386158403263398536375017221519, 9.220822426325422108086155984248

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