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Properties

Label 2-303450-1.1-c1-0-14
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 − 5·11-s + 12-s − 13-s + 14-s + 16-s − 18-s + 4·19-s − 21-s + 5·22-s − 24-s + 26-s + 27-s − 28-s + 6·29-s − 3·31-s − 32-s − 5·33-s + 36-s − 12·37-s − 4·38-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 + 0.917·19-s − 0.218·21-s + 1.06·22-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.538·31-s − 0.176·32-s − 0.870·33-s + 1/6·36-s − 1.97·37-s − 0.648·38-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

\(L(1)\) \(\approx\) \(0.9543110009\)
\(L(\frac12)\) \(\approx\) \(0.9543110009\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−12.61610648565877, −12.21327908645906, −11.86190482896266, −11.12028477946376, −10.62510395187234, −10.41807038901577, −9.896450366083290, −9.405602524614638, −9.092280441921335, −8.437985217627490, −8.120611381029165, −7.599963237644030, −7.261498778531472, −6.796810400238986, −6.162707228715519, −5.593287883125786, −5.109490245719991, −4.656057685633585, −3.875487774004955, −3.215199901165041, −2.922931061375123, −2.380528477086154, −1.781532020362428, −1.093489235927236, −0.2930279360723141, 0.2930279360723141, 1.093489235927236, 1.781532020362428, 2.380528477086154, 2.922931061375123, 3.215199901165041, 3.875487774004955, 4.656057685633585, 5.109490245719991, 5.593287883125786, 6.162707228715519, 6.796810400238986, 7.261498778531472, 7.599963237644030, 8.120611381029165, 8.437985217627490, 9.092280441921335, 9.405602524614638, 9.896450366083290, 10.41807038901577, 10.62510395187234, 11.12028477946376, 11.86190482896266, 12.21327908645906, 12.61610648565877

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