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Properties

Label 2-1519-1519.960-c0-0-2
Degree $2$
Conductor $1519$
Sign $-0.572 + 0.820i$
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  + (0.400 + 0.193i)2-s + (−0.499 − 0.626i)4-s + (−0.277 − 1.21i)5-s + (0.623 − 0.781i)7-s + (−0.178 − 0.781i)8-s + (−0.900 + 0.433i)9-s + (0.123 − 0.541i)10-s + (0.400 − 0.193i)14-s + (−0.0990 + 0.433i)16-s − 0.445·18-s − 1.80·19-s + (−0.623 + 0.781i)20-s + (−0.499 + 0.240i)25-s − 0.801·28-s + 31-s + (−0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (0.400 + 0.193i)2-s + (−0.499 − 0.626i)4-s + (−0.277 − 1.21i)5-s + (0.623 − 0.781i)7-s + (−0.178 − 0.781i)8-s + (−0.900 + 0.433i)9-s + (0.123 − 0.541i)10-s + (0.400 − 0.193i)14-s + (−0.0990 + 0.433i)16-s − 0.445·18-s − 1.80·19-s + (−0.623 + 0.781i)20-s + (−0.499 + 0.240i)25-s − 0.801·28-s + 31-s + (−0.623 + 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $-0.572 + 0.820i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (960, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ -0.572 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9038185873\)
\(L(\frac12)\) \(\approx\) \(0.9038185873\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
3 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−9.205943536903099741010555950800, −8.492580847596297008700618618092, −8.088611872652104368256600990978, −6.82748892083244868770210882171, −5.91704073256349018241706191895, −5.01881837064990257255096461116, −4.57000422743649007792630371073, −3.75675746811805536154519884846, −1.99863588433615834744707568327, −0.63192379204025569377786035867, 2.36698595611411684646533441317, 2.93999823199766499976562205728, 3.92851930594286327274508246365, 4.82523862917812214104177339479, 5.88208335866010617666171448831, 6.58264651198463890074238061506, 7.65974899192229372854371892290, 8.484181498202097522516476092459, 8.819619513869921314683475228780, 10.00855042290029773931008510675

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