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Properties

Label 2-1519-1519.1425-c0-0-0
Degree $2$
Conductor $1519$
Sign $0.708 - 0.705i$
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  + (−0.205 − 0.140i)2-s + (−0.342 − 0.873i)4-s + (−0.0476 + 0.0146i)5-s + (−0.0249 + 0.999i)7-s + (−0.107 + 0.469i)8-s + (0.0747 + 0.997i)9-s + (0.0118 + 0.00365i)10-s + (0.145 − 0.201i)14-s + (−0.599 + 0.556i)16-s + (0.124 − 0.215i)18-s + (0.583 + 1.01i)19-s + (0.0291 + 0.0365i)20-s + (−0.824 + 0.561i)25-s + (0.881 − 0.320i)28-s + (−0.5 + 0.866i)31-s + (0.677 − 0.102i)32-s + ⋯
L(s)  = 1  + (−0.205 − 0.140i)2-s + (−0.342 − 0.873i)4-s + (−0.0476 + 0.0146i)5-s + (−0.0249 + 0.999i)7-s + (−0.107 + 0.469i)8-s + (0.0747 + 0.997i)9-s + (0.0118 + 0.00365i)10-s + (0.145 − 0.201i)14-s + (−0.599 + 0.556i)16-s + (0.124 − 0.215i)18-s + (0.583 + 1.01i)19-s + (0.0291 + 0.0365i)20-s + (−0.824 + 0.561i)25-s + (0.881 − 0.320i)28-s + (−0.5 + 0.866i)31-s + (0.677 − 0.102i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (1425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ 0.708 - 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8076436656\)
\(L(\frac12)\) \(\approx\) \(0.8076436656\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.0249 - 0.999i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.205 + 0.140i)T + (0.365 + 0.930i)T^{2} \)
3 \( 1 + (-0.0747 - 0.997i)T^{2} \)
5 \( 1 + (0.0476 - 0.0146i)T + (0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.583 - 1.01i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (-0.141 + 0.621i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-1.57 - 1.07i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-1.33 - 0.411i)T + (0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.14 + 1.44i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + 1.32T + T^{2} \)
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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

−9.675404214079928540064226512768, −9.112390900094967320047418149192, −8.263150614878783563456472519719, −7.53303808955381570455141812458, −6.32096219425931149204059242736, −5.48357721354498304221741210985, −5.09353657375719399620108529450, −3.83748149615600344862292295608, −2.45878919749269044982564315872, −1.58091466059671635022455444974, 0.73468549618586767410355084280, 2.64835323758241340906051003061, 3.80807275432816174593195991025, 4.15483940206550009922149784132, 5.41955539951749231625703142445, 6.66724098456457010777806061999, 7.14557341333714264065233782505, 7.944398599551933080702445250743, 8.745079182961323667788218800209, 9.550696417358480642886202279164

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