Properties

Label 2-95370-1.1-c1-0-76
Degree $2$
Conductor $95370$
Sign $-1$
Analytic cond. $761.533$
Root an. cond. $27.5958$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 2·14-s + 15-s + 16-s − 18-s + 20-s + 2·21-s + 22-s + 4·23-s − 24-s + 25-s + 27-s + 2·28-s − 2·29-s − 30-s − 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 0.436·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95370\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(761.533\)
Root analytic conductor: \(27.5958\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{95370} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 95370,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.28871140460968, −13.44353972084008, −13.12134230313038, −12.70592834946349, −12.00682357149163, −11.38658734713367, −11.09444113667857, −10.56524068986207, −9.915062894618959, −9.632433113012775, −9.064452433270756, −8.430263313133153, −8.264428959194023, −7.619642816036750, −6.968408509941284, −6.783232937436458, −5.799655458701693, −5.426023463844463, −4.794531997210196, −4.153578275017175, −3.402475363478120, −2.820734725338237, −2.194971913658804, −1.620437871520514, −1.053006552819734, 0, 1.053006552819734, 1.620437871520514, 2.194971913658804, 2.820734725338237, 3.402475363478120, 4.153578275017175, 4.794531997210196, 5.426023463844463, 5.799655458701693, 6.783232937436458, 6.968408509941284, 7.619642816036750, 8.264428959194023, 8.430263313133153, 9.064452433270756, 9.632433113012775, 9.915062894618959, 10.56524068986207, 11.09444113667857, 11.38658734713367, 12.00682357149163, 12.70592834946349, 13.12134230313038, 13.44353972084008, 14.28871140460968

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