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Properties

Label 2-1520-5.4-c1-0-29
Degree $2$
Conductor $1520$
Sign $0.309 + 0.950i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  − 2.84i·3-s + (0.691 + 2.12i)5-s + 0.145i·7-s − 5.10·9-s + 5.71·11-s − 5.24i·13-s + (6.05 − 1.96i)15-s + 7.15i·17-s + 19-s + 0.412·21-s − 0.622i·23-s + (−4.04 + 2.94i)25-s + 6.00i·27-s + 5.46·29-s + 3.77·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (0.309 + 0.950i)5-s + 0.0548i·7-s − 1.70·9-s + 1.72·11-s − 1.45i·13-s + (1.56 − 0.508i)15-s + 1.73i·17-s + 0.229·19-s + 0.0901·21-s − 0.129i·23-s + (−0.808 + 0.588i)25-s + 1.15i·27-s + 1.01·29-s + 0.678·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.962373912\)
\(L(\frac12)\) \(\approx\) \(1.962373912\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.691 - 2.12i)T \)
19 \( 1 - T \)
good3 \( 1 + 2.84iT - 3T^{2} \)
7 \( 1 - 0.145iT - 7T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + 5.24iT - 13T^{2} \)
17 \( 1 - 7.15iT - 17T^{2} \)
23 \( 1 + 0.622iT - 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 - 3.77T + 31T^{2} \)
37 \( 1 + 5.03iT - 37T^{2} \)
41 \( 1 - 5.77T + 41T^{2} \)
43 \( 1 + 3.32iT - 43T^{2} \)
47 \( 1 + 5.85iT - 47T^{2} \)
53 \( 1 - 6.97iT - 53T^{2} \)
59 \( 1 - 9.09T + 59T^{2} \)
61 \( 1 + 8.16T + 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + 2.41T + 71T^{2} \)
73 \( 1 + 7.44iT - 73T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + 2.17iT - 83T^{2} \)
89 \( 1 + 3.90T + 89T^{2} \)
97 \( 1 - 4.98iT - 97T^{2} \)
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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

−9.116874323884118177732056541440, −8.287086326222519113161816348626, −7.60760274764492804996292881954, −6.81132953531931430246541974082, −6.20654788642185527333516964842, −5.72846058891521949810104398215, −3.96817626991333269538831862234, −2.98443584051759968539176604089, −1.96345648673209389123102334946, −0.975240058292870222364100414435, 1.18480890710461835710753949727, 2.79602950245416530076975536493, 4.11978257236709199208711908587, 4.39913364856039961940055294990, 5.20191899164261512619694374484, 6.22497232793593668368454619660, 7.09949181505727591634125011624, 8.587509997347669803833019339851, 8.994478917215631643926262168220, 9.688145478827945684007999974394

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