Properties

Label 2-303450-1.1-c1-0-68
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 + 3·11-s − 12-s − 4·13-s + 14-s + 16-s − 18-s + 8·19-s + 21-s − 3·22-s + 3·23-s + 24-s + 4·26-s − 27-s − 28-s + 6·29-s − 2·31-s − 32-s − 3·33-s + 36-s − 5·37-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 + 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.218·21-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s − 0.821·37-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.915990360\)
\(L(\frac12)\) \(\approx\) \(1.915990360\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.49722847112193, −12.05635654067409, −11.69766334682171, −11.48300029322697, −10.77727013876023, −10.27239532692222, −9.791417235413809, −9.727627100278310, −8.898091237576555, −8.787284828405798, −7.975133671199312, −7.483014880813965, −7.051829001086682, −6.704477428034431, −6.300513315544436, −5.540835894924870, −5.025883816987942, −4.915072613417977, −3.786603683393269, −3.579047617343809, −2.865350470213738, −2.254370693973345, −1.622412399640873, −0.8289410419056247, −0.5881895450229414, 0.5881895450229414, 0.8289410419056247, 1.622412399640873, 2.254370693973345, 2.865350470213738, 3.579047617343809, 3.786603683393269, 4.915072613417977, 5.025883816987942, 5.540835894924870, 6.300513315544436, 6.704477428034431, 7.051829001086682, 7.483014880813965, 7.975133671199312, 8.787284828405798, 8.898091237576555, 9.727627100278310, 9.791417235413809, 10.27239532692222, 10.77727013876023, 11.48300029322697, 11.69766334682171, 12.05635654067409, 12.49722847112193

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