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Properties

Label 2-303450-1.1-c1-0-11
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 $0$

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 − 7·13-s − 14-s + 16-s − 18-s + 8·19-s − 21-s − 2·22-s − 5·23-s + 24-s + 7·26-s − 27-s + 28-s − 9·29-s − 31-s − 32-s − 2·33-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 + 0.603·11-s − 0.288·12-s − 1.94·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.218·21-s − 0.426·22-s − 1.04·23-s + 0.204·24-s + 1.37·26-s − 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.179·31-s − 0.176·32-s − 0.348·33-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: $1$
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: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5849364005\)
\(L(\frac12)\) \(\approx\) \(0.5849364005\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
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

−12.43657555709515, −12.02016701582816, −11.78180104825547, −11.34387321686412, −10.94814649943606, −10.19407101764201, −9.819368007715971, −9.654026210563936, −9.180228074429027, −8.497137546648519, −7.851395530008343, −7.694156663875493, −7.022089984765091, −6.830218393301683, −6.186631799284602, −5.397494125910988, −5.200873516545924, −4.853946095935601, −3.790271211430975, −3.676949341621323, −2.795410959141957, −2.106957243179675, −1.749784387181197, −1.029875869966713, −0.2597621947613809, 0.2597621947613809, 1.029875869966713, 1.749784387181197, 2.106957243179675, 2.795410959141957, 3.676949341621323, 3.790271211430975, 4.853946095935601, 5.200873516545924, 5.397494125910988, 6.186631799284602, 6.830218393301683, 7.022089984765091, 7.694156663875493, 7.851395530008343, 8.497137546648519, 9.180228074429027, 9.654026210563936, 9.819368007715971, 10.19407101764201, 10.94814649943606, 11.34387321686412, 11.78180104825547, 12.02016701582816, 12.43657555709515

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