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Properties

Label 2-303450-1.1-c1-0-135
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 − 5·13-s + 14-s + 16-s + 18-s − 4·19-s − 21-s + 5·22-s − 2·23-s − 24-s − 5·26-s − 27-s + 28-s − 4·29-s + 7·31-s + 32-s − 5·33-s + 36-s + 6·37-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 − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.218·21-s + 1.06·22-s − 0.417·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.188·28-s − 0.742·29-s + 1.25·31-s + 0.176·32-s − 0.870·33-s + 1/6·36-s + 0.986·37-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 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.77268554560917, −12.41237052224881, −12.01210498560260, −11.56668428220313, −11.31613926686375, −10.76158039598963, −10.19258794760970, −9.777711997270467, −9.328705931750422, −8.817807478389344, −8.179864111809300, −7.545316522324205, −7.378682944052555, −6.572655069123274, −6.325563888790574, −5.924015625987577, −5.301404245223429, −4.651719707944087, −4.325832448080881, −4.138864767640724, −3.175560244839845, −2.772448196250920, −1.939500162229734, −1.620302422732049, −0.8235738832902540, 0, 0.8235738832902540, 1.620302422732049, 1.939500162229734, 2.772448196250920, 3.175560244839845, 4.138864767640724, 4.325832448080881, 4.651719707944087, 5.301404245223429, 5.924015625987577, 6.325563888790574, 6.572655069123274, 7.378682944052555, 7.545316522324205, 8.179864111809300, 8.817807478389344, 9.328705931750422, 9.777711997270467, 10.19258794760970, 10.76158039598963, 11.31613926686375, 11.56668428220313, 12.01210498560260, 12.41237052224881, 12.77268554560917

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