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Properties

Label 2-1519-217.123-c0-0-1
Degree $2$
Conductor $1519$
Sign $0.386 - 0.922i$
Analytic cond. $0.758079$
Root an. cond. $0.870677$
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)5-s + 8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)31-s + (−0.499 + 0.866i)38-s + (0.499 + 0.866i)40-s − 41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s + (0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)31-s + (−0.499 + 0.866i)38-s + (0.499 + 0.866i)40-s − 41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s + (0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676892460\)
\(L(\frac12)\) \(\approx\) \(1.676892460\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
31 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.02615308589040029173012744557, −8.902264164128842793892442054404, −8.036722901853991563032676656790, −7.00041372742349385275271944046, −6.61150482595742757980419471094, −5.80131568641837758434104961341, −5.18874105438898270256957796624, −3.91456708499095300231461329588, −3.00813035561280530498233893967, −1.66360725819855976484094091015, 1.44245599534422989729290570143, 2.42332882596807080427336740261, 3.33053878687794730611647045656, 4.57702365163549852515790679274, 5.04539794613731082652622753114, 5.97700974673574873453828731241, 7.24490233020250215802836124123, 7.952905096785869610920164318591, 8.833577430373471413404445248365, 9.587564340529044906171767578752

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