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Properties

Label 2-2352-147.71-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.159 + 0.987i$
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  + (0.222 − 0.974i)3-s + (0.900 − 0.433i)7-s + (−0.900 − 0.433i)9-s + (0.400 − 0.193i)13-s + 1.80·19-s + (−0.222 − 0.974i)21-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)27-s − 1.24·31-s + (0.777 + 0.974i)37-s + (−0.0990 − 0.433i)39-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (0.400 − 1.75i)57-s + (0.777 + 0.974i)61-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (0.900 − 0.433i)7-s + (−0.900 − 0.433i)9-s + (0.400 − 0.193i)13-s + 1.80·19-s + (−0.222 − 0.974i)21-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)27-s − 1.24·31-s + (0.777 + 0.974i)37-s + (−0.0990 − 0.433i)39-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (0.400 − 1.75i)57-s + (0.777 + 0.974i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406345381\)
\(L(\frac12)\) \(\approx\) \(1.406345381\)
\(L(1)\) 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 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
good5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - 1.80T + T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + 1.24T + T^{2} \)
37 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 - 0.445T + T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
79 \( 1 + 1.24T + T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−8.783989415361646006821334703160, −8.106930863491083949935221568592, −7.47399053301820012490542664393, −6.93966951809792354459734269174, −5.81124601957785218342148342279, −5.26958765735723791034762720426, −4.04239241367830015674202825615, −3.13534854562305443186481070582, −1.97985926202554947572110468195, −1.04735070427353098130690145083, 1.60978297879763721002819278916, 2.80430915870790385829449677143, 3.70855311562524481993833244826, 4.54889818446441486585548787877, 5.42422557559695772474310089525, 5.84597354071186378795429344976, 7.25395619712546835699550796438, 7.942178302914808465062671092540, 8.631993060473502087124286707320, 9.494898705493439159626706264499

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