Properties

Label 2-5520-92.91-c1-0-82
Degree $2$
Conductor $5520$
Sign $-0.759 + 0.650i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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 i·5-s + 2.30·7-s − 9-s + 0.945·11-s − 5.38·13-s − 15-s + 6.56i·17-s + 2.72·19-s − 2.30i·21-s + (0.880 + 4.71i)23-s − 25-s + i·27-s − 6.26·29-s − 5.84i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s + 0.870·7-s − 0.333·9-s + 0.284·11-s − 1.49·13-s − 0.258·15-s + 1.59i·17-s + 0.626·19-s − 0.502i·21-s + (0.183 + 0.982i)23-s − 0.200·25-s + 0.192i·27-s − 1.16·29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240034795\)
\(L(\frac12)\) \(\approx\) \(1.240034795\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
23 \( 1 + (-0.880 - 4.71i)T \)
good7 \( 1 - 2.30T + 7T^{2} \)
11 \( 1 - 0.945T + 11T^{2} \)
13 \( 1 + 5.38T + 13T^{2} \)
17 \( 1 - 6.56iT - 17T^{2} \)
19 \( 1 - 2.72T + 19T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + 6.23iT - 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 8.79T + 43T^{2} \)
47 \( 1 + 9.81iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 + 5.84iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 6.05T + 67T^{2} \)
71 \( 1 + 9.51iT - 71T^{2} \)
73 \( 1 - 5.20T + 73T^{2} \)
79 \( 1 + 2.04T + 79T^{2} \)
83 \( 1 + 3.94T + 83T^{2} \)
89 \( 1 + 2.77iT - 89T^{2} \)
97 \( 1 + 9.24iT - 97T^{2} \)
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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

−7.80759682440678629395658309288, −7.39325123084236762906127694551, −6.49028110320547015744752790070, −5.57362383184254235708023936277, −5.14859668489458673441837604943, −4.20180885189638258827436735634, −3.41784787261276304659144719599, −2.05621757983320660918469516800, −1.70233852669278532340557303652, −0.32317890641237859629158327884, 1.19142328926741830686266380352, 2.55820533938923316454901634034, 2.94293539665229835649042790983, 4.19899020700744073760891776004, 4.82065346186837151319497724057, 5.29781771006857089479042894223, 6.27566690559991299185033937639, 7.33978042633663393039485832234, 7.43309555716533039964503598686, 8.464070544829935937234632245176

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