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Properties

Label 2-303450-1.1-c1-0-50
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 − 4·11-s − 12-s − 6·13-s + 14-s + 16-s − 18-s + 4·19-s + 21-s + 4·22-s + 24-s + 6·26-s − 27-s − 28-s + 2·29-s + 8·31-s − 32-s + 4·33-s + 36-s − 10·37-s − 4·38-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 − 1.66·13-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 + 0.204·24-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s − 1.64·37-s − 0.648·38-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
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

−12.70473889121044, −12.30405706557848, −12.00697487942574, −11.57155653456696, −11.02012377656805, −10.35539547272693, −10.19237750971557, −9.900643452928587, −9.300272259114932, −8.841803729802736, −8.190183676710602, −7.737088073288753, −7.389879598041736, −6.954721133953786, −6.381545951690181, −5.960777141107930, −5.136984377195216, −5.053252148632275, −4.544773326816619, −3.618691202777646, −2.975457378579920, −2.701349694643305, −1.984582682656056, −1.354562492590431, −0.5002870408672872, 0, 0.5002870408672872, 1.354562492590431, 1.984582682656056, 2.701349694643305, 2.975457378579920, 3.618691202777646, 4.544773326816619, 5.053252148632275, 5.136984377195216, 5.960777141107930, 6.381545951690181, 6.954721133953786, 7.389879598041736, 7.737088073288753, 8.190183676710602, 8.841803729802736, 9.300272259114932, 9.900643452928587, 10.19237750971557, 10.35539547272693, 11.02012377656805, 11.57155653456696, 12.00697487942574, 12.30405706557848, 12.70473889121044

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