Properties

Label 2-303450-1.1-c1-0-123
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 − 12-s − 4·13-s − 14-s + 16-s + 18-s + 2·19-s + 21-s + 9·23-s − 24-s − 4·26-s − 27-s − 28-s − 2·31-s + 32-s + 36-s + 7·37-s + 2·38-s + 4·39-s − 3·41-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 − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.218·21-s + 1.87·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.359·31-s + 0.176·32-s + 1/6·36-s + 1.15·37-s + 0.324·38-s + 0.640·39-s − 0.468·41-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: 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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.74273684892944, −12.68003339771013, −11.86390168307007, −11.74177317825198, −11.06732948056748, −10.86304697547110, −10.18205249414136, −9.738058772781786, −9.386935576322124, −8.882528415418147, −7.978672304594026, −7.845157851161692, −7.022705298945039, −6.759384680000894, −6.499787948756623, −5.578456501627772, −5.354430556956488, −4.921863802638904, −4.401419356524310, −3.849483096858263, −3.111207576514727, −2.864837166131039, −2.153622008140987, −1.438889167038556, −0.7934489309612372, 0, 0.7934489309612372, 1.438889167038556, 2.153622008140987, 2.864837166131039, 3.111207576514727, 3.849483096858263, 4.401419356524310, 4.921863802638904, 5.354430556956488, 5.578456501627772, 6.499787948756623, 6.759384680000894, 7.022705298945039, 7.845157851161692, 7.978672304594026, 8.882528415418147, 9.386935576322124, 9.738058772781786, 10.18205249414136, 10.86304697547110, 11.06732948056748, 11.74177317825198, 11.86390168307007, 12.68003339771013, 12.74273684892944

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