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Properties

Label 2-2352-12.11-c1-0-59
Degree $2$
Conductor $2352$
Sign $0.791 + 0.611i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (1.60 − 0.656i)3-s − 0.670i·5-s + (2.13 − 2.10i)9-s + 5.24·11-s + 2·13-s + (−0.440 − 1.07i)15-s + 4.21i·17-s − 1.31i·19-s − 3.20·23-s + 4.54·25-s + (2.04 − 4.77i)27-s − 6.22i·29-s + 1.73i·31-s + (8.41 − 3.44i)33-s − 6.27·37-s + ⋯
L(s)  = 1  + (0.925 − 0.379i)3-s − 0.300i·5-s + (0.712 − 0.701i)9-s + 1.58·11-s + 0.554·13-s + (−0.113 − 0.277i)15-s + 1.02i·17-s − 0.301i·19-s − 0.668·23-s + 0.909·25-s + (0.393 − 0.919i)27-s − 1.15i·29-s + 0.311i·31-s + (1.46 − 0.600i)33-s − 1.03·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.948792237\)
\(L(\frac12)\) \(\approx\) \(2.948792237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.60 + 0.656i)T \)
7 \( 1 \)
good5 \( 1 + 0.670iT - 5T^{2} \)
11 \( 1 - 5.24T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 4.21iT - 17T^{2} \)
19 \( 1 + 1.31iT - 19T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + 6.22iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 6.27T + 37T^{2} \)
41 \( 1 - 9.76iT - 41T^{2} \)
43 \( 1 - 9.55iT - 43T^{2} \)
47 \( 1 - 9.61T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 7.57T + 59T^{2} \)
61 \( 1 + 3.72T + 61T^{2} \)
67 \( 1 + 2.98iT - 67T^{2} \)
71 \( 1 - 4.08T + 71T^{2} \)
73 \( 1 - 0.274T + 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 - 2.04T + 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + 3.27T + 97T^{2} \)
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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

−8.790864080448055366562984045931, −8.308534100537726714949874415917, −7.47644264546844788163902875054, −6.46248517187055970599778305573, −6.17337013502097832159003121285, −4.65708697834537301689288092308, −3.93543763601327036330714970531, −3.18873377947031595613302080698, −1.90464197320030347645127945763, −1.10540343999697545810086469301, 1.27058218406049754620280536191, 2.37339766978233231975006619838, 3.46579602862620555326337417145, 3.95829470782805761475996060006, 4.95636069145486619614107333196, 5.99317948562611764455631097427, 7.02284911280753072693232860497, 7.37192029609681098995053357787, 8.691602712325433065197390056358, 8.878042566459032790157281697869

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