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Properties

Label 2-303450-1.1-c1-0-109
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 + 13-s + 14-s + 16-s + 18-s − 6·19-s − 21-s − 5·22-s + 6·23-s − 24-s + 26-s − 27-s + 28-s − 6·29-s − 4·31-s + 32-s + 5·33-s + 36-s + 11·37-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.218·21-s − 1.06·22-s + 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.870·33-s + 1/6·36-s + 1.80·37-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 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.99918710154967, −12.61045524910735, −12.04576335200617, −11.36772215599511, −11.15072477371620, −10.83109210595059, −10.26756774968469, −9.997469113583817, −9.083043467459311, −8.844966889048667, −8.140817287795805, −7.681133923028359, −7.237303877692536, −6.893490144413105, −6.046600203676493, −5.771921801492348, −5.459241189254281, −4.763196069686510, −4.374448275036165, −4.018915782328878, −3.127434057222421, −2.722087243318393, −2.140629490814011, −1.548632112485370, −0.7444127269932999, 0, 0.7444127269932999, 1.548632112485370, 2.140629490814011, 2.722087243318393, 3.127434057222421, 4.018915782328878, 4.374448275036165, 4.763196069686510, 5.459241189254281, 5.771921801492348, 6.046600203676493, 6.893490144413105, 7.237303877692536, 7.681133923028359, 8.140817287795805, 8.844966889048667, 9.083043467459311, 9.997469113583817, 10.26756774968469, 10.83109210595059, 11.15072477371620, 11.36772215599511, 12.04576335200617, 12.61045524910735, 12.99918710154967

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