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Properties

Label 2-2352-588.131-c0-0-1
Degree $2$
Conductor $2352$
Sign $0.414 - 0.910i$
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  + (0.0747 + 0.997i)3-s + (−0.365 − 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (0.365 − 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.733 + 1.26i)31-s + (1.40 + 1.29i)37-s + (−1.12 + 1.64i)39-s + (−0.129 − 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (−0.365 − 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (0.365 − 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.733 + 1.26i)31-s + (1.40 + 1.29i)37-s + (−1.12 + 1.64i)39-s + (−0.129 − 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196633458\)
\(L(\frac12)\) \(\approx\) \(1.196633458\)
\(L(1)\) 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 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.365 + 0.930i)T \)
good5 \( 1 + (0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (-1.55 - 1.24i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.826 - 0.563i)T^{2} \)
19 \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (0.129 + 0.268i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-1.32 + 1.42i)T + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.975 - 0.563i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.548 + 0.215i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.955 - 0.294i)T^{2} \)
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

−9.372979773526865232691993714268, −8.697890532929081371022316231148, −7.934982169534776921934857864315, −6.81442238223357303437580656416, −6.30682087713071789877617699055, −5.21740780169573002470051005159, −4.35372227282489023068631962850, −3.73863415065891219819199194144, −2.93986864003179983734631360634, −1.33353574528254700881290457374, 0.939974867931265672778694289678, 2.25797877292481663635480540608, 3.04018017361886409418382249292, 4.01526766339652420239267546966, 5.55090916884848747043357297209, 5.92358883337273585328934351918, 6.52129801033701493957555784770, 7.74999540767805500272254009009, 8.125756414206403420489034300093, 8.842032885373518418435764945074

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