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Properties

Label 2-1560-1560.1403-c0-0-3
Degree $2$
Conductor $1560$
Sign $-0.229 - 0.973i$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
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  + (0.707 + 0.707i)2-s + (−0.5 − 0.866i)3-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + (0.258 − 0.965i)6-s + (−1.22 + 1.22i)7-s + (−0.707 + 0.707i)8-s + (−0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.866 − 0.500i)12-s + (0.707 + 0.707i)13-s − 1.73·14-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (1.36 + 1.36i)17-s + (−0.965 + 0.258i)18-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.5 − 0.866i)3-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + (0.258 − 0.965i)6-s + (−1.22 + 1.22i)7-s + (−0.707 + 0.707i)8-s + (−0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.866 − 0.500i)12-s + (0.707 + 0.707i)13-s − 1.73·14-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (1.36 + 1.36i)17-s + (−0.965 + 0.258i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9452363476\)
\(L(\frac12)\) \(\approx\) \(0.9452363476\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
47 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.511396837612868670601114868504, −8.776510363841565971432602227119, −8.204202673094879301240036280317, −7.33372601345514370876202692081, −6.34062973198834430759545862384, −5.84427966691479019290462172013, −5.31717009219395415875327398768, −4.04894625129553888989873840331, −3.10273314721915476193497053929, −1.73310815068788685387437250656, 0.63611146449134114639580549659, 2.84717386715660152968024220475, 3.60006914549651792146201059434, 3.88176596921532049422334476752, 5.26437283049197508100930530735, 5.91723307944549878465354523991, 6.83927408686370419818644468014, 7.44666370852734065482839517704, 9.096017393764000589123701308390, 9.814850345516566645352689907122

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