Properties

Label 2-2520-5.4-c1-0-30
Degree $2$
Conductor $2520$
Sign $-0.590 + 0.807i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  + (−1.32 + 1.80i)5-s i·7-s − 2.48·11-s + 4.15i·13-s − 5.76i·17-s + 1.60·19-s + 7.28i·23-s + (−1.51 − 4.76i)25-s + 1.45·29-s − 2.24·31-s + (1.80 + 1.32i)35-s − 6i·37-s − 11.2·41-s + 5.28i·43-s − 3.45i·47-s + ⋯
L(s)  = 1  + (−0.590 + 0.807i)5-s − 0.377i·7-s − 0.749·11-s + 1.15i·13-s − 1.39i·17-s + 0.369·19-s + 1.51i·23-s + (−0.303 − 0.952i)25-s + 0.270·29-s − 0.404·31-s + (0.305 + 0.223i)35-s − 0.986i·37-s − 1.76·41-s + 0.805i·43-s − 0.503i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.590 + 0.807i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.590 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3298576360\)
\(L(\frac12)\) \(\approx\) \(0.3298576360\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.32 - 1.80i)T \)
7 \( 1 + iT \)
good11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 - 4.15iT - 13T^{2} \)
17 \( 1 + 5.76iT - 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 - 7.28iT - 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 5.28iT - 43T^{2} \)
47 \( 1 + 3.45iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 - 5.35T + 61T^{2} \)
67 \( 1 + 7.52iT - 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 7.28iT - 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 2.73iT - 97T^{2} \)
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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

−8.598137893553393304077211697005, −7.62336933741006483103263908720, −7.23156705299691680905703718289, −6.56073240447391171011035319895, −5.43888733996543168059939791893, −4.65858354855003893101126239931, −3.66231536049378816869838850409, −2.96440028540264394697127446812, −1.81732444337859393287451200494, −0.11401191124837913187552011752, 1.23616658409360555570265336177, 2.57674070095778207485601798572, 3.52523163992515637473555000619, 4.48972827846643921426873194349, 5.26749221670850524785313010134, 5.91260491716056674272459690736, 6.94572518461306347545729417735, 7.957681013899598350533164758404, 8.338707343852302555290565357833, 8.897293298471173502900242323856

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