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Properties

Label 2-303240-1.1-c1-0-12
Degree $2$
Conductor $303240$
Sign $1$
Analytic cond. $2421.38$
Root an. cond. $49.2075$
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  + 3-s + 5-s − 7-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s − 21-s + 25-s + 27-s + 10·29-s − 4·33-s − 35-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s − 8·47-s + 49-s + 2·51-s − 6·53-s − 4·55-s + 4·59-s − 10·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.696·33-s − 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2421.38\)
Root analytic conductor: \(49.2075\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.407947345\)
\(L(\frac12)\) \(\approx\) \(2.407947345\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.63860902746211, −12.44056069390216, −11.83916288055052, −11.22760322648579, −10.67819162853298, −10.39286112761461, −9.836111950746196, −9.647541958364962, −8.867135882716307, −8.581084350843631, −7.994760130039590, −7.783825360015604, −7.013114522518161, −6.616945014782276, −6.167838415904460, −5.477871185736738, −5.198039752406887, −4.515072900301745, −4.053006935079471, −3.259708465754701, −2.905259655031148, −2.577712540510836, −1.688543623560627, −1.295525532330525, −0.3869499584167043, 0.3869499584167043, 1.295525532330525, 1.688543623560627, 2.577712540510836, 2.905259655031148, 3.259708465754701, 4.053006935079471, 4.515072900301745, 5.198039752406887, 5.477871185736738, 6.167838415904460, 6.616945014782276, 7.013114522518161, 7.783825360015604, 7.994760130039590, 8.581084350843631, 8.867135882716307, 9.647541958364962, 9.836111950746196, 10.39286112761461, 10.67819162853298, 11.22760322648579, 11.83916288055052, 12.44056069390216, 12.63860902746211

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