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Properties

Label 2-4560-76.75-c1-0-62
Degree $2$
Conductor $4560$
Sign $0.527 + 0.849i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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  + 3-s + 5-s + 0.624i·7-s + 9-s − 4.32i·11-s + 0.624i·13-s + 15-s − 4.41·17-s + (2.29 + 3.70i)19-s + 0.624i·21-s − 1.92i·23-s + 25-s + 27-s − 8.16i·29-s + 0.700·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.235i·7-s + 0.333·9-s − 1.30i·11-s + 0.173i·13-s + 0.258·15-s − 1.07·17-s + (0.527 + 0.849i)19-s + 0.136i·21-s − 0.400i·23-s + 0.200·25-s + 0.192·27-s − 1.51i·29-s + 0.125·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.527 + 0.849i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 0.527 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.408907050\)
\(L(\frac12)\) \(\approx\) \(2.408907050\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 + (-2.29 - 3.70i)T \)
good7 \( 1 - 0.624iT - 7T^{2} \)
11 \( 1 + 4.32iT - 11T^{2} \)
13 \( 1 - 0.624iT - 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
23 \( 1 + 1.92iT - 23T^{2} \)
29 \( 1 + 8.16iT - 29T^{2} \)
31 \( 1 - 0.700T + 31T^{2} \)
37 \( 1 + 2.40iT - 37T^{2} \)
41 \( 1 + 7.89iT - 41T^{2} \)
43 \( 1 + 2.40iT - 43T^{2} \)
47 \( 1 - 1.64iT - 47T^{2} \)
53 \( 1 + 1.92iT - 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 6.31T + 73T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 + 4.23iT - 83T^{2} \)
89 \( 1 + 3.61iT - 89T^{2} \)
97 \( 1 + 2.94iT - 97T^{2} \)
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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

−8.464881704656446212768986531256, −7.55966932086506406833203336896, −6.74753507938084447576263212578, −5.95937315904030988614933357911, −5.44624710794125625760310594600, −4.29545759190883916924411178965, −3.63507533309830281167738766545, −2.64945514652933965653788654693, −1.97097929146070074038713796714, −0.63334779707832422623855757176, 1.19758518214071097704278029252, 2.17462958026142884452628282929, 2.90170614978822426749132787852, 3.93877105368231593638522753713, 4.74535899484357862297987623940, 5.31550330103741866599796011472, 6.53639989480340343010100528324, 7.00490657866320479372057076998, 7.63049937887675706646191473073, 8.557231912862540832555819636977

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