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Properties

Label 2-665-19.7-c1-0-26
Degree $2$
Conductor $665$
Sign $0.238 + 0.971i$
Analytic cond. $5.31005$
Root an. cond. $2.30435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.675 − 1.17i)3-s + (1 − 1.73i)4-s + (0.5 + 0.866i)5-s + 7-s + (0.586 − 1.01i)9-s + 5.17·11-s − 2.70·12-s + (−1 + 1.73i)13-s + (0.675 − 1.17i)15-s + (−1.99 − 3.46i)16-s + (−0.351 − 0.609i)17-s + (1.91 + 3.91i)19-s + 1.99·20-s + (−0.675 − 1.17i)21-s + (2.76 − 4.78i)23-s + ⋯
L(s)  = 1  + (−0.390 − 0.675i)3-s + (0.5 − 0.866i)4-s + (0.223 + 0.387i)5-s + 0.377·7-s + (0.195 − 0.338i)9-s + 1.55·11-s − 0.780·12-s + (−0.277 + 0.480i)13-s + (0.174 − 0.302i)15-s + (−0.499 − 0.866i)16-s + (−0.0853 − 0.147i)17-s + (0.438 + 0.898i)19-s + 0.447·20-s + (−0.147 − 0.255i)21-s + (0.575 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(665\)    =    \(5 \cdot 7 \cdot 19\)
Sign: $0.238 + 0.971i$
Analytic conductor: \(5.31005\)
Root analytic conductor: \(2.30435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{665} (596, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 665,\ (\ :1/2),\ 0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35581 - 1.06319i\)
\(L(\frac12)\) \(\approx\) \(1.35581 - 1.06319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
19 \( 1 + (-1.91 - 3.91i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.675 + 1.17i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.351 + 0.609i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.76 + 4.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.85 - 3.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.82T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 + (4.08 + 7.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 - 6.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.52 + 4.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.82 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.413 - 0.716i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.93 - 3.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-4.26 + 7.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.87 - 8.44i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.35737301052584921855618713844, −9.598524599242189191034964555744, −8.729091166602708194184200045242, −7.25132223526524975865937485074, −6.72833433249717229857829809153, −6.12217054302690278921154839684, −5.04132452468578451201922401357, −3.70281177899608783973360560580, −2.02994869980025784886523812867, −1.12174822512070078898066327636, 1.64414928021479718420897213953, 3.18964867804754293255362322908, 4.26807698214335605336871959369, 5.05335202002362261970458141249, 6.25889617449958587570387990645, 7.21682324352628313418196066245, 8.084858272923133752884982285331, 9.071634547941162552709949619402, 9.782489682407259864431691330564, 10.85916950659726165772674206196

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