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Properties

Label 2-303450-1.1-c1-0-7
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 − 6·11-s − 12-s − 13-s + 14-s + 16-s − 18-s − 4·19-s + 21-s + 6·22-s + 3·23-s + 24-s + 26-s − 27-s − 28-s − 3·29-s − 5·31-s − 32-s + 6·33-s + 36-s + 10·37-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 − 1.80·11-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 + 1.27·22-s + 0.625·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + 1.64·37-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

\(L(1)\) \(\approx\) \(0.3998162242\)
\(L(\frac12)\) \(\approx\) \(0.3998162242\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.65001847443107, −12.27273692721606, −11.66293878539781, −11.04678372794323, −10.88629619329869, −10.41012621762849, −10.00877838080591, −9.491282352237031, −9.128156286476885, −8.389292697827047, −8.095289901300170, −7.586337608311205, −7.127343335442131, −6.651201085834693, −6.180420522487135, −5.503960460086181, −5.268920186900588, −4.718458067737247, −3.996325118204422, −3.443831475492084, −2.696879109907702, −2.359869922945168, −1.755851801107971, −0.8690387221822015, −0.2392953185551831, 0.2392953185551831, 0.8690387221822015, 1.755851801107971, 2.359869922945168, 2.696879109907702, 3.443831475492084, 3.996325118204422, 4.718458067737247, 5.268920186900588, 5.503960460086181, 6.180420522487135, 6.651201085834693, 7.127343335442131, 7.586337608311205, 8.095289901300170, 8.389292697827047, 9.128156286476885, 9.491282352237031, 10.00877838080591, 10.41012621762849, 10.88629619329869, 11.04678372794323, 11.66293878539781, 12.27273692721606, 12.65001847443107

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