Properties

Label 2-3120-5.4-c1-0-31
Degree $2$
Conductor $3120$
Sign $0.783 - 0.621i$
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  + i·3-s + (−1.75 + 1.38i)5-s − 9-s − 1.50·11-s i·13-s + (−1.38 − 1.75i)15-s − 2.72i·17-s + 0.726·19-s − 4.72i·23-s + (1.14 − 4.86i)25-s i·27-s + 7.55·29-s + 3.00·31-s − 1.50i·33-s − 5.00i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.783 + 0.621i)5-s − 0.333·9-s − 0.453·11-s − 0.277i·13-s + (−0.358 − 0.452i)15-s − 0.661i·17-s + 0.166·19-s − 0.985i·23-s + (0.228 − 0.973i)25-s − 0.192i·27-s + 1.40·29-s + 0.540·31-s − 0.261i·33-s − 0.823i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.783 - 0.621i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.369375433\)
\(L(\frac12)\) \(\approx\) \(1.369375433\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (1.75 - 1.38i)T \)
13 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
17 \( 1 + 2.72iT - 17T^{2} \)
19 \( 1 - 0.726T + 19T^{2} \)
23 \( 1 + 4.72iT - 23T^{2} \)
29 \( 1 - 7.55T + 29T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 + 5.00iT - 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 - 2.72iT - 43T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 - 7.55iT - 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 6.28T + 61T^{2} \)
67 \( 1 - 12.5iT - 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 5.27T + 79T^{2} \)
83 \( 1 - 7.78iT - 83T^{2} \)
89 \( 1 + 1.78T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.742315897698293989163884126062, −7.977617394882667166333712971625, −7.40000020896567877945642469242, −6.51805259157415515274548803139, −5.74867537595910101144280752122, −4.66238542577320898374108318458, −4.22658777644963277224711219020, −3.02697237492869481377911624690, −2.61763409108892445073203585862, −0.69706277352882374359123859339, 0.71317089395745856639903613525, 1.79708542088910533601733190947, 2.99047286123964886112421061733, 3.88912970204085948270739178323, 4.77699109396742859052047280204, 5.52062340413919473029141197597, 6.44936986249266500869688333585, 7.21233845404225744217441788930, 7.955173554820651309057013525535, 8.421302608766991071759371647430

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