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Properties

Label 2-1560-1560.269-c0-0-1
Degree $2$
Conductor $1560$
Sign $0.711 - 0.702i$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 0.999·12-s + 13-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 0.999·12-s + 13-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8480379111\)
\(L(\frac12)\) \(\approx\) \(0.8480379111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.438827029812069215815704600489, −9.092061012535080725187990643948, −8.453831279452819355182988911683, −7.09288555621271866295315504473, −6.30387252797722128642919068015, −5.85500097202114891120571040823, −4.96422001781070184180479282518, −4.17175986930326810390147019023, −2.74337734980586080443724486004, −0.959785707704206291136636425719, 1.55355723330266909215166577630, 1.84679325572080217614422133075, 3.25831024479075637765049845308, 4.44925817453683798031657553177, 5.48682223299315912144021539152, 6.46721336865464664793760250788, 7.02329607225904035155392442246, 8.177708439657629270415185107944, 8.787274121113389895342471685884, 9.563889744768469472636304181367

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