Properties

Label 2-2352-12.11-c1-0-38
Degree $2$
Conductor $2352$
Sign $0.998 - 0.0581i$
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  + (0.777 − 1.54i)3-s + 3.40i·5-s + (−1.79 − 2.40i)9-s + 3.94·13-s + (5.26 + 2.64i)15-s − 6.09i·17-s + 4.38i·19-s + 8.33·23-s − 6.58·25-s + (−5.11 + 0.901i)27-s − 3.80i·29-s + 4.89i·31-s − 7.58·37-s + (3.06 − 6.10i)39-s + 0.710i·41-s + ⋯
L(s)  = 1  + (0.448 − 0.893i)3-s + 1.52i·5-s + (−0.597 − 0.802i)9-s + 1.09·13-s + (1.36 + 0.683i)15-s − 1.47i·17-s + 1.00i·19-s + 1.73·23-s − 1.31·25-s + (−0.984 + 0.173i)27-s − 0.707i·29-s + 0.879i·31-s − 1.24·37-s + (0.491 − 0.978i)39-s + 0.110i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 - 0.0581i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.998 - 0.0581i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.196865831\)
\(L(\frac12)\) \(\approx\) \(2.196865831\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.777 + 1.54i)T \)
7 \( 1 \)
good5 \( 1 - 3.40iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 + 6.09iT - 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 - 8.33T + 23T^{2} \)
29 \( 1 + 3.80iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 - 0.710iT - 41T^{2} \)
43 \( 1 - 9.66iT - 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 - 4.66T + 59T^{2} \)
61 \( 1 - 1.70T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 6.09iT - 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−8.915020526996671595057735228354, −8.082698287736334683426096518475, −7.29428501842082560694488124275, −6.80897967659725086771244741467, −6.19073022886682181482033307874, −5.22497615651142982016049568103, −3.71255370833893814805062186162, −3.09963897467688643330563869286, −2.34863916346160983455493667225, −1.05266019353768213869728442869, 0.900020064271311023397419323673, 2.11992196419696910433883540208, 3.51284144407523929894149377649, 4.08543161808776288631345972313, 5.05957634105159861689712346813, 5.45186912845435539189902717596, 6.57804882388653714269515183296, 7.74409120573568090373776108939, 8.624830237326801319491791287571, 8.862500830259485860516727279305

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