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Properties

Label 2-2352-21.20-c1-0-36
Degree $2$
Conductor $2352$
Sign $0.999 + 0.0212i$
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  + (−1.33 + 1.10i)3-s + 0.145·5-s + (0.554 − 2.94i)9-s + 2.46i·11-s − 2.04i·13-s + (−0.193 + 0.160i)15-s + 1.75·17-s + 4.25i·19-s − 8.61i·23-s − 4.97·25-s + (2.52 + 4.54i)27-s − 7.08i·29-s + 3.60i·31-s + (−2.73 − 3.29i)33-s + 5.86·37-s + ⋯
L(s)  = 1  + (−0.769 + 0.638i)3-s + 0.0649·5-s + (0.184 − 0.982i)9-s + 0.744i·11-s − 0.566i·13-s + (−0.0500 + 0.0414i)15-s + 0.426·17-s + 0.975i·19-s − 1.79i·23-s − 0.995·25-s + (0.485 + 0.874i)27-s − 1.31i·29-s + 0.646i·31-s + (−0.475 − 0.573i)33-s + 0.964·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.999 + 0.0212i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.999 + 0.0212i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231402837\)
\(L(\frac12)\) \(\approx\) \(1.231402837\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 1.10i)T \)
7 \( 1 \)
good5 \( 1 - 0.145T + 5T^{2} \)
11 \( 1 - 2.46iT - 11T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 + 8.61iT - 23T^{2} \)
29 \( 1 + 7.08iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 5.86T + 37T^{2} \)
41 \( 1 - 5.33T + 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 + 5.19iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 5.38iT - 61T^{2} \)
67 \( 1 - 5.14T + 67T^{2} \)
71 \( 1 + 7.79iT - 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 5.72T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 - 8.68T + 89T^{2} \)
97 \( 1 - 6.65iT - 97T^{2} \)
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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

−9.228541169476167828697187343139, −8.116277244878133068348455089215, −7.52762290573828590247260667748, −6.28329658024437140212366277181, −6.02083256257480182610864518774, −4.90924732705496081441290360024, −4.31887366185480610890960752058, −3.38642322823194929991213315339, −2.13868133204960656589432737325, −0.63373484043106877796165691784, 0.882079108738514981639563985546, 1.95992545846675926763256217062, 3.14640376195606951398841691729, 4.25850802687768512236545248576, 5.25555959746871659153927837679, 5.87325991575400642458961365480, 6.59229099684637924642550410589, 7.49885034034477332318039480872, 7.929705835622462663043066625222, 9.101711477920817709910153578392

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