↑

Properties

Label 2-2352-12.11-c1-0-40
Degree $2$
Conductor $2352$
Sign $0.957 + 0.288i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 − 0.5i)3-s + 3.31i·5-s + (2.5 + 1.65i)9-s + 3.31·11-s − 4·13-s + (1.65 − 5.5i)15-s − 3.31i·17-s − 7i·19-s + 3.31·23-s − 6·25-s + (−3.31 − 4i)27-s − 6.63i·29-s − 3i·31-s + (−5.5 − 1.65i)33-s − 37-s + ⋯
L(s)  = 1  + (−0.957 − 0.288i)3-s + 1.48i·5-s + (0.833 + 0.552i)9-s + 1.00·11-s − 1.10·13-s + (0.428 − 1.42i)15-s − 0.804i·17-s − 1.60i·19-s + 0.691·23-s − 1.20·25-s + (−0.638 − 0.769i)27-s − 1.23i·29-s − 0.538i·31-s + (−0.957 − 0.288i)33-s − 0.164·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.957 + 0.288i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.164151697\)
\(L(\frac12)\) \(\approx\) \(1.164151697\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.5i)T \)
7 \( 1 \)
good5 \( 1 - 3.31iT - 5T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 - 3.31T + 23T^{2} \)
29 \( 1 + 6.63iT - 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 6.63iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 9.94T + 47T^{2} \)
53 \( 1 - 3.31iT - 53T^{2} \)
59 \( 1 + 3.31T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 9iT - 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 3.31iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.251601827517411294704776635951, −7.79886223233091892133691366831, −7.13165241318093080136592252292, −6.74344131836764239232338439347, −6.04841372145232596017626697262, −4.99688988708014104975088302908, −4.25812783298571084350987260458, −2.97252362417964098296084082983, −2.22570325336136179062630474824, −0.59744382209633611690000424288, 0.929981671462054324155053037832, 1.77135205231088166297709692721, 3.67325216289455862058976769645, 4.29162284538368623740324640065, 5.20264489347812509052820192454, 5.59953796382860905098672878526, 6.62529923020960553339147426611, 7.38665540776207782472469515572, 8.460903018679712962479657092053, 9.029827819316140349954629722756

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