Properties

Degree $2$
Conductor $303450$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

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 − 4·11-s − 12-s + 2·13-s − 14-s + 16-s − 18-s − 4·19-s − 21-s + 4·22-s − 8·23-s + 24-s − 2·26-s − 27-s + 28-s − 6·29-s + 8·31-s − 32-s + 4·33-s + 36-s − 2·37-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 − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s − 0.328·37-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: $1$
Motivic weight: \(1\)
Character: $\chi_{303450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.99835198310320, −12.62211275223012, −12.11443477102443, −11.63904315730817, −11.31443900883356, −10.65615927127912, −10.37682717222538, −10.21959892175990, −9.459483121623567, −8.979023507140706, −8.456337120032204, −8.088813520622629, −7.485074334666660, −7.384732141551858, −6.468865426609667, −6.119101203084282, −5.701587948476680, −5.265029810396375, −4.441189565799292, −4.191894273437887, −3.521690431809905, −2.554766643241515, −2.442029755046897, −1.580973545446008, −1.116575514655914, 0, 0, 1.116575514655914, 1.580973545446008, 2.442029755046897, 2.554766643241515, 3.521690431809905, 4.191894273437887, 4.441189565799292, 5.265029810396375, 5.701587948476680, 6.119101203084282, 6.468865426609667, 7.384732141551858, 7.485074334666660, 8.088813520622629, 8.456337120032204, 8.979023507140706, 9.459483121623567, 10.21959892175990, 10.37682717222538, 10.65615927127912, 11.31443900883356, 11.63904315730817, 12.11443477102443, 12.62211275223012, 12.99835198310320

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