Properties

Label 2-41070-1.1-c1-0-3
Degree $2$
Conductor $41070$
Sign $1$
Analytic cond. $327.945$
Root an. cond. $18.1092$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 12-s − 2·13-s − 4·14-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s + 24-s + 25-s − 2·26-s + 27-s − 4·28-s + 6·29-s + 30-s − 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.182·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41070\)    =    \(2 \cdot 3 \cdot 5 \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(327.945\)
Root analytic conductor: \(18.1092\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 41070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.306341395\)
\(L(\frac12)\) \(\approx\) \(3.306341395\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
37 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.69524190471727, −14.09110584012878, −13.69304754286180, −13.26698441879388, −12.67848170172788, −12.53829263142101, −11.71820011663418, −11.16099819465517, −10.47084271348970, −9.950654412083061, −9.545246853378187, −8.992310010233821, −8.496278894752842, −7.588442765248437, −7.038400974279028, −6.675625417492124, −6.101241678312556, −5.457460316167117, −4.796607264067746, −4.158511576407779, −3.434743403919719, −2.989456944997064, −2.382499605960354, −1.721576403737266, −0.5330110556768413, 0.5330110556768413, 1.721576403737266, 2.382499605960354, 2.989456944997064, 3.434743403919719, 4.158511576407779, 4.796607264067746, 5.457460316167117, 6.101241678312556, 6.675625417492124, 7.038400974279028, 7.588442765248437, 8.496278894752842, 8.992310010233821, 9.545246853378187, 9.950654412083061, 10.47084271348970, 11.16099819465517, 11.71820011663418, 12.53829263142101, 12.67848170172788, 13.26698441879388, 13.69304754286180, 14.09110584012878, 14.69524190471727

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