Properties

Label 2-303450-1.1-c1-0-133
Degree $2$
Conductor $303450$
Sign $-1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 − 6-s − 7-s − 8-s + 9-s − 5·11-s + 12-s + 2·13-s + 14-s + 16-s − 18-s − 2·19-s − 21-s + 5·22-s + 4·23-s − 24-s − 2·26-s + 27-s − 28-s + 9·29-s + 9·31-s − 32-s − 5·33-s + 36-s − 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 1.06·22-s + 0.834·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + 1.61·31-s − 0.176·32-s − 0.870·33-s + 1/6·36-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.90136223045014, −12.40900879378702, −12.10395195505046, −11.42352733339177, −10.88596725721787, −10.43443697909456, −10.21597568561342, −9.804784840797269, −9.073274095295849, −8.717961948468613, −8.327159210591495, −7.994285666024622, −7.405996452552817, −6.894106734803798, −6.512390124475896, −5.997217588429705, −5.202304251055629, −4.996890519311320, −4.208547448753554, −3.608109332049488, −3.053579173494942, −2.449085904726406, −2.351471188444723, −1.269031361002141, −0.8226646744819130, 0, 0.8226646744819130, 1.269031361002141, 2.351471188444723, 2.449085904726406, 3.053579173494942, 3.608109332049488, 4.208547448753554, 4.996890519311320, 5.202304251055629, 5.997217588429705, 6.512390124475896, 6.894106734803798, 7.405996452552817, 7.994285666024622, 8.327159210591495, 8.717961948468613, 9.073274095295849, 9.804784840797269, 10.21597568561342, 10.43443697909456, 10.88596725721787, 11.42352733339177, 12.10395195505046, 12.40900879378702, 12.90136223045014

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