Properties

Label 2-5520-92.91-c1-0-6
Degree $2$
Conductor $5520$
Sign $-0.927 + 0.373i$
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 − 1.53·7-s − 9-s + 3.80·11-s + 2.39·13-s − 15-s + 2.51i·17-s − 3.13·19-s − 1.53i·21-s + (−3.77 − 2.95i)23-s − 25-s i·27-s − 1.76·29-s − 3.33i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s − 0.579·7-s − 0.333·9-s + 1.14·11-s + 0.663·13-s − 0.258·15-s + 0.610i·17-s − 0.719·19-s − 0.334i·21-s + (−0.787 − 0.616i)23-s − 0.200·25-s − 0.192i·27-s − 0.328·29-s − 0.599i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.927 + 0.373i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.927 + 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4576446804\)
\(L(\frac12)\) \(\approx\) \(0.4576446804\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.77 + 2.95i)T \)
good7 \( 1 + 1.53T + 7T^{2} \)
11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 2.51iT - 17T^{2} \)
19 \( 1 + 3.13T + 19T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
31 \( 1 + 3.33iT - 31T^{2} \)
37 \( 1 - 3.84iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 0.753T + 43T^{2} \)
47 \( 1 - 9.86iT - 47T^{2} \)
53 \( 1 + 5.02iT - 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + 1.33T + 67T^{2} \)
71 \( 1 - 7.54iT - 71T^{2} \)
73 \( 1 - 8.62T + 73T^{2} \)
79 \( 1 - 2.23T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 1.26iT - 97T^{2} \)
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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

−8.441808192663913521390055721639, −8.117384340970694513497151884748, −6.81982798307638692630139953209, −6.44931019314806707049875870704, −5.87379757622366134984264621826, −4.81218556847413078801324826836, −3.88530431542795455076431532995, −3.60651871153936367639074783053, −2.51502031584048633036552044867, −1.45224496784654582478276248846, 0.11839501194736505142265799426, 1.32507404910949881016292905094, 2.08904109418698117249590188340, 3.36449872904604690494273936825, 3.86157490326189496515759481512, 4.89939885576858291923006314870, 5.71662355699637806198620204822, 6.49065162115392003582476829603, 6.84530549737978834170406793282, 7.78807214039710078317304532490

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