Properties

Label 2-303450-1.1-c1-0-126
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 + 2·11-s − 12-s + 14-s + 16-s − 18-s + 19-s + 21-s − 2·22-s + 24-s − 27-s − 28-s − 29-s + 8·31-s − 32-s − 2·33-s + 36-s + 2·37-s − 38-s − 2·41-s − 42-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 + 0.603·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.218·21-s − 0.426·22-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + 0.328·37-s − 0.162·38-s − 0.312·41-s − 0.154·42-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: $\chi_{303450} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−12.88906606217271, −12.22855445832998, −11.89401181932252, −11.55158906562540, −11.11150259361870, −10.53747122640910, −10.06142268643654, −9.835123389883542, −9.256695651003016, −8.837936203748012, −8.257336701390581, −7.905237995114159, −7.233663536446359, −6.826956936180309, −6.413916868000123, −6.020762121097214, −5.412607783451434, −4.872988560958841, −4.354088154264665, −3.626157733512810, −3.298786464544512, −2.457128403470922, −2.024086324360006, −1.164229383677689, −0.8028774596763561, 0, 0.8028774596763561, 1.164229383677689, 2.024086324360006, 2.457128403470922, 3.298786464544512, 3.626157733512810, 4.354088154264665, 4.872988560958841, 5.412607783451434, 6.020762121097214, 6.413916868000123, 6.826956936180309, 7.233663536446359, 7.905237995114159, 8.257336701390581, 8.837936203748012, 9.256695651003016, 9.835123389883542, 10.06142268643654, 10.53747122640910, 11.11150259361870, 11.55158906562540, 11.89401181932252, 12.22855445832998, 12.88906606217271

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