Properties

Label 2-1560-5.4-c1-0-30
Degree $2$
Conductor $1560$
Sign $-0.0332 + 0.999i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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 + (2.23 + 0.0743i)5-s − 2.82i·7-s − 9-s − 6.46·11-s i·13-s + (−0.0743 + 2.23i)15-s − 3.49i·17-s + 2.21·19-s + 2.82·21-s − 7.14i·23-s + (4.98 + 0.332i)25-s i·27-s − 4.82·29-s − 9.65·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.999 + 0.0332i)5-s − 1.06i·7-s − 0.333·9-s − 1.95·11-s − 0.277i·13-s + (−0.0191 + 0.577i)15-s − 0.847i·17-s + 0.507·19-s + 0.617·21-s − 1.49i·23-s + (0.997 + 0.0664i)25-s − 0.192i·27-s − 0.896·29-s − 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.0332 + 0.999i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ -0.0332 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.214095030\)
\(L(\frac12)\) \(\approx\) \(1.214095030\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0743i)T \)
13 \( 1 + iT \)
good7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 6.46T + 11T^{2} \)
17 \( 1 + 3.49iT - 17T^{2} \)
19 \( 1 - 2.21T + 19T^{2} \)
23 \( 1 + 7.14iT - 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + 6.93iT - 37T^{2} \)
41 \( 1 + 0.148T + 41T^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + 6.79iT - 47T^{2} \)
53 \( 1 - 1.70iT - 53T^{2} \)
59 \( 1 - 6.46T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 - 7.70iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 5.76iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 2.27iT - 83T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 - 8.11iT - 97T^{2} \)
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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

−9.436886369193311339099402378386, −8.516206492921603424957241366998, −7.53872311023313026958267554001, −6.96554965401148065254067719497, −5.59911907864675778761732323322, −5.27955782105145920150578864203, −4.24393876829061025245587016028, −3.07260984292618042604436863267, −2.20788435781368937605566624201, −0.43335557391396205171008749605, 1.68517479021493962117793564076, 2.39140435046119567956236934495, 3.37300552935516608115193733045, 5.16780986400323261628063807146, 5.49263140406077109917206421401, 6.20637412147887014761255354386, 7.35594834609925223118855382252, 7.977782396483636045206026416943, 8.915488093266831323749003710341, 9.515929403888089945935758713177

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