Properties

Label 2-2352-1.1-c3-0-19
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·5-s + 9·9-s − 62·11-s + 62·13-s − 12·15-s − 84·17-s + 100·19-s + 42·23-s − 109·25-s − 27·27-s − 10·29-s − 48·31-s + 186·33-s − 246·37-s − 186·39-s + 248·41-s − 68·43-s + 36·45-s + 324·47-s + 252·51-s + 258·53-s − 248·55-s − 300·57-s + 120·59-s − 622·61-s + 248·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.357·5-s + 1/3·9-s − 1.69·11-s + 1.32·13-s − 0.206·15-s − 1.19·17-s + 1.20·19-s + 0.380·23-s − 0.871·25-s − 0.192·27-s − 0.0640·29-s − 0.278·31-s + 0.981·33-s − 1.09·37-s − 0.763·39-s + 0.944·41-s − 0.241·43-s + 0.119·45-s + 1.00·47-s + 0.691·51-s + 0.668·53-s − 0.608·55-s − 0.697·57-s + 0.264·59-s − 1.30·61-s + 0.473·65-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: $1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.427700646\)
\(L(\frac12)\) \(\approx\) \(1.427700646\)
\(L(\frac{5}{2})\) 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 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 62 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 - 42 T + p^{3} T^{2} \)
29 \( 1 + 10 T + p^{3} T^{2} \)
31 \( 1 + 48 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 - 248 T + p^{3} T^{2} \)
43 \( 1 + 68 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 - 258 T + p^{3} T^{2} \)
59 \( 1 - 120 T + p^{3} T^{2} \)
61 \( 1 + 622 T + p^{3} T^{2} \)
67 \( 1 + 904 T + p^{3} T^{2} \)
71 \( 1 - 678 T + p^{3} T^{2} \)
73 \( 1 - 642 T + p^{3} T^{2} \)
79 \( 1 + 740 T + p^{3} T^{2} \)
83 \( 1 - 468 T + p^{3} T^{2} \)
89 \( 1 + 200 T + p^{3} T^{2} \)
97 \( 1 - 1266 T + p^{3} T^{2} \)
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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

−8.668874478444222324675055800060, −7.77904173129166353273208368779, −7.11725125920832870184523017025, −6.11058717768176017929820293210, −5.56326333387366993036216960487, −4.84199907198275710178546328785, −3.79787626953185884783762783834, −2.76059097826502589963903009519, −1.74062537040470956194289552420, −0.54439763031612193366950759789, 0.54439763031612193366950759789, 1.74062537040470956194289552420, 2.76059097826502589963903009519, 3.79787626953185884783762783834, 4.84199907198275710178546328785, 5.56326333387366993036216960487, 6.11058717768176017929820293210, 7.11725125920832870184523017025, 7.77904173129166353273208368779, 8.668874478444222324675055800060

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