Properties

Label 2-2352-12.11-c1-0-68
Degree $2$
Conductor $2352$
Sign $0.329 + 0.944i$
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  + (1.70 + 0.323i)3-s − 1.55i·5-s + (2.79 + 1.09i)9-s − 2.53·13-s + (0.502 − 2.64i)15-s − 4.33i·17-s − 6.83i·19-s − 3.80·23-s + 2.58·25-s + (4.39 + 2.77i)27-s − 8.33i·29-s + 4.89i·31-s + 1.58·37-s + (−4.31 − 0.818i)39-s − 7.44i·41-s + ⋯
L(s)  = 1  + (0.982 + 0.186i)3-s − 0.695i·5-s + (0.930 + 0.366i)9-s − 0.702·13-s + (0.129 − 0.683i)15-s − 1.05i·17-s − 1.56i·19-s − 0.794·23-s + 0.516·25-s + (0.845 + 0.533i)27-s − 1.54i·29-s + 0.879i·31-s + 0.260·37-s + (−0.690 − 0.131i)39-s − 1.16i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.329 + 0.944i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326964801\)
\(L(\frac12)\) \(\approx\) \(2.326964801\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.323i)T \)
7 \( 1 \)
good5 \( 1 + 1.55iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2.53T + 13T^{2} \)
17 \( 1 + 4.33iT - 17T^{2} \)
19 \( 1 + 6.83iT - 19T^{2} \)
23 \( 1 + 3.80T + 23T^{2} \)
29 \( 1 + 8.33iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 1.58T + 37T^{2} \)
41 \( 1 + 7.44iT - 41T^{2} \)
43 \( 1 + 6.20iT - 43T^{2} \)
47 \( 1 + 5.38T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 8.19T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 4.82T + 83T^{2} \)
89 \( 1 + 4.33iT - 89T^{2} \)
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

−8.834740752828440960292984838401, −8.242761540445103312941461415785, −7.31887462199056344876980401136, −6.82631192015691729782654896872, −5.42966447494770659242717338679, −4.74621372862435785130181442555, −4.04459518865902132337967175876, −2.85502780801859980337389406750, −2.18163599066935038818789561074, −0.69173339477106989740752220423, 1.50507522720360483509853255266, 2.40742514637745053384540082463, 3.39019437622004944252738651733, 3.99952419429271343112706968292, 5.14600471242184446367329507139, 6.30125886935291410948641284755, 6.81386232618610164599568951206, 7.910338868549873558524737436221, 8.075776680033342679941905563429, 9.110308962676599777638448874554

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