Properties

Degree $2$
Conductor $2352$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 15·5-s + 9·9-s + 9·11-s − 88·13-s − 45·15-s − 84·17-s − 104·19-s + 84·23-s + 100·25-s + 27·27-s + 51·29-s − 185·31-s + 27·33-s + 44·37-s − 264·39-s − 168·41-s − 326·43-s − 135·45-s + 138·47-s − 252·51-s + 639·53-s − 135·55-s − 312·57-s − 159·59-s + 722·61-s + 1.32e3·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.246·11-s − 1.87·13-s − 0.774·15-s − 1.19·17-s − 1.25·19-s + 0.761·23-s + 4/5·25-s + 0.192·27-s + 0.326·29-s − 1.07·31-s + 0.142·33-s + 0.195·37-s − 1.08·39-s − 0.639·41-s − 1.15·43-s − 0.447·45-s + 0.428·47-s − 0.691·51-s + 1.65·53-s − 0.330·55-s − 0.725·57-s − 0.350·59-s + 1.51·61-s + 2.51·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8373211712\)
\(L(\frac12)\) \(\approx\) \(0.8373211712\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p T \)
7 \( 1 \)
good5 \( 1 + 3 p T + p^{3} T^{2} \)
11 \( 1 - 9 T + p^{3} T^{2} \)
13 \( 1 + 88 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 + 104 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 - 51 T + p^{3} T^{2} \)
31 \( 1 + 185 T + p^{3} T^{2} \)
37 \( 1 - 44 T + p^{3} T^{2} \)
41 \( 1 + 168 T + p^{3} T^{2} \)
43 \( 1 + 326 T + p^{3} T^{2} \)
47 \( 1 - 138 T + p^{3} T^{2} \)
53 \( 1 - 639 T + p^{3} T^{2} \)
59 \( 1 + 159 T + p^{3} T^{2} \)
61 \( 1 - 722 T + p^{3} T^{2} \)
67 \( 1 - 166 T + p^{3} T^{2} \)
71 \( 1 + 1086 T + p^{3} T^{2} \)
73 \( 1 - 218 T + p^{3} T^{2} \)
79 \( 1 - 583 T + p^{3} T^{2} \)
83 \( 1 - 597 T + p^{3} T^{2} \)
89 \( 1 + 1038 T + p^{3} T^{2} \)
97 \( 1 + 169 T + p^{3} T^{2} \)
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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

−8.598888575735632628722983513671, −7.896894365920455243857616307243, −7.09027071411951814820804381897, −6.74648021870752000068215955424, −5.23170839607373734031207925857, −4.44584047169741010531083964907, −3.87814763004806321810686816076, −2.80589273592558127370164639484, −2.00158110782737273438785954798, −0.37292761984696651787224378544, 0.37292761984696651787224378544, 2.00158110782737273438785954798, 2.80589273592558127370164639484, 3.87814763004806321810686816076, 4.44584047169741010531083964907, 5.23170839607373734031207925857, 6.74648021870752000068215955424, 7.09027071411951814820804381897, 7.896894365920455243857616307243, 8.598888575735632628722983513671

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