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Properties

Label 2-3120-12.11-c1-0-44
Degree $2$
Conductor $3120$
Sign $0.787 + 0.615i$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  + (−0.648 − 1.60i)3-s + i·5-s − 3.10i·7-s + (−2.15 + 2.08i)9-s + 3.67·11-s + 13-s + (1.60 − 0.648i)15-s − 3.22i·17-s + 7.68i·19-s + (−4.99 + 2.01i)21-s + 8.17·23-s − 25-s + (4.74 + 2.11i)27-s + 1.87i·29-s + 5.90i·31-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)3-s + 0.447i·5-s − 1.17i·7-s + (−0.719 + 0.694i)9-s + 1.10·11-s + 0.277·13-s + (0.414 − 0.167i)15-s − 0.781i·17-s + 1.76i·19-s + (−1.08 + 0.439i)21-s + 1.70·23-s − 0.200·25-s + (0.913 + 0.407i)27-s + 0.347i·29-s + 1.06i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.787 + 0.615i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.787 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738255508\)
\(L(\frac12)\) \(\approx\) \(1.738255508\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.648 + 1.60i)T \)
5 \( 1 - iT \)
13 \( 1 - T \)
good7 \( 1 + 3.10iT - 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
17 \( 1 + 3.22iT - 17T^{2} \)
19 \( 1 - 7.68iT - 19T^{2} \)
23 \( 1 - 8.17T + 23T^{2} \)
29 \( 1 - 1.87iT - 29T^{2} \)
31 \( 1 - 5.90iT - 31T^{2} \)
37 \( 1 + 2.51T + 37T^{2} \)
41 \( 1 + 2.35iT - 41T^{2} \)
43 \( 1 - 4.16iT - 43T^{2} \)
47 \( 1 - 3.77T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 2.06T + 59T^{2} \)
61 \( 1 - 5.31T + 61T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 + 9.03T + 71T^{2} \)
73 \( 1 - 4.78T + 73T^{2} \)
79 \( 1 - 7.58iT - 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 - 9.91iT - 89T^{2} \)
97 \( 1 + 4.27T + 97T^{2} \)
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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

−8.503637472009206508610007088013, −7.58474065372423158032289544396, −7.08164527240029038753541139124, −6.57298906418838559775063801348, −5.75291829897394100076272383563, −4.79716326901821464229159175645, −3.77571525682223535213307981655, −3.01927570068721934882196593710, −1.61617416160849896981120452879, −0.911305815070815394182835905266, 0.803419854140755402891678388394, 2.27939847949858410835386312012, 3.28966440492483075186157023006, 4.19382619495684545507038019800, 4.94270872721360462902450575705, 5.62124474841842220692005129413, 6.34141453892807038321895524885, 7.09890102692226839316568999111, 8.487099367768948096461950629621, 8.899143780218893063665497775250

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