Properties

Label 2-6035-1.1-c1-0-276
Degree $2$
Conductor $6035$
Sign $-1$
Analytic cond. $48.1897$
Root an. cond. $6.94188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.12·2-s + 0.383·3-s − 0.742·4-s + 5-s − 0.430·6-s + 1.34·7-s + 3.07·8-s − 2.85·9-s − 1.12·10-s + 1.85·11-s − 0.285·12-s − 0.128·13-s − 1.50·14-s + 0.383·15-s − 1.96·16-s + 17-s + 3.19·18-s + 1.93·19-s − 0.742·20-s + 0.516·21-s − 2.08·22-s + 0.872·23-s + 1.18·24-s + 25-s + 0.144·26-s − 2.24·27-s − 0.998·28-s + ⋯
L(s)  = 1  − 0.792·2-s + 0.221·3-s − 0.371·4-s + 0.447·5-s − 0.175·6-s + 0.508·7-s + 1.08·8-s − 0.950·9-s − 0.354·10-s + 0.560·11-s − 0.0822·12-s − 0.0356·13-s − 0.403·14-s + 0.0991·15-s − 0.490·16-s + 0.242·17-s + 0.754·18-s + 0.443·19-s − 0.166·20-s + 0.112·21-s − 0.444·22-s + 0.181·23-s + 0.240·24-s + 0.200·25-s + 0.0282·26-s − 0.432·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6035\)    =    \(5 \cdot 17 \cdot 71\)
Sign: $-1$
Analytic conductor: \(48.1897\)
Root analytic conductor: \(6.94188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6035,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
17 \( 1 - T \)
71 \( 1 + T \)
good2 \( 1 + 1.12T + 2T^{2} \)
3 \( 1 - 0.383T + 3T^{2} \)
7 \( 1 - 1.34T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 + 0.128T + 13T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 - 0.872T + 23T^{2} \)
29 \( 1 + 8.06T + 29T^{2} \)
31 \( 1 + 0.702T + 31T^{2} \)
37 \( 1 + 3.28T + 37T^{2} \)
41 \( 1 + 0.772T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 9.60T + 47T^{2} \)
53 \( 1 + 2.79T + 53T^{2} \)
59 \( 1 + 7.92T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 1.83T + 67T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 0.0156T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−7.82607072743879342983484001359, −7.31705921172961404201628748928, −6.35348072425502012284484746075, −5.47624712133499469670517651017, −4.99333117067013987570544032668, −3.97902547331589964086035435599, −3.19693475516495074442141990315, −2.03647047254962616645114179932, −1.28761142360305209041041679560, 0, 1.28761142360305209041041679560, 2.03647047254962616645114179932, 3.19693475516495074442141990315, 3.97902547331589964086035435599, 4.99333117067013987570544032668, 5.47624712133499469670517651017, 6.35348072425502012284484746075, 7.31705921172961404201628748928, 7.82607072743879342983484001359

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