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Properties

Label 2-3120-3120.1949-c0-0-6
Degree $2$
Conductor $3120$
Sign $-0.707 + 0.707i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.382 − 0.923i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)8-s − 1.00i·9-s + i·10-s + (1.30 − 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + (−0.923 − 0.382i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.382 − 0.923i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)8-s − 1.00i·9-s + i·10-s + (1.30 − 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + (−0.923 − 0.382i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076777252\)
\(L(\frac12)\) \(\approx\) \(1.076777252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
61 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 0.765iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
89 \( 1 - 0.765iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.676919870415639600106965795011, −7.964564524973476803308576393349, −7.57276586455896026103621294746, −6.28150156683061180547341770358, −6.07230946080914383830679835286, −4.38213218014956255582200443092, −3.51823802208411885779075521542, −2.91659369541828432984680787429, −1.44225761114754454211218776793, −0.927900830250554856697380126940, 1.72320411827727060778914378297, 2.47798858666492894679992701157, 3.73855943209806311825619114021, 4.23914268041554433813010861094, 5.47185009499512398444829180504, 6.47264940883523013441965368204, 7.15404848833341444782722967168, 7.53989170022209833029579039021, 8.728468876788522857419441271851, 8.959444924538432664851873074873

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