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Properties

Label 2-303450-1.1-c1-0-107
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 − 3·11-s − 12-s − 2·13-s − 14-s + 16-s + 18-s + 2·19-s + 21-s − 3·22-s − 24-s − 2·26-s − 27-s − 28-s + 3·29-s + 5·31-s + 32-s + 3·33-s + 36-s − 2·37-s + 2·38-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 − 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.639·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 0.328·37-s + 0.324·38-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.97166847644236, −12.50341903746713, −11.98493498247441, −11.66988689264309, −11.17679051191512, −10.59476479033706, −10.33538157683935, −9.748389786847315, −9.484479089931559, −8.664169345231606, −8.212845668494997, −7.542911533667479, −7.390798515388322, −6.597507268124442, −6.381757145470051, −5.742091695665674, −5.316501496114470, −4.840072099242152, −4.469321266161697, −3.827587269459673, −3.163274036534542, −2.761216187136499, −2.216865048683182, −1.461513360376037, −0.7392427665594293, 0, 0.7392427665594293, 1.461513360376037, 2.216865048683182, 2.761216187136499, 3.163274036534542, 3.827587269459673, 4.469321266161697, 4.840072099242152, 5.316501496114470, 5.742091695665674, 6.381757145470051, 6.597507268124442, 7.390798515388322, 7.542911533667479, 8.212845668494997, 8.664169345231606, 9.484479089931559, 9.748389786847315, 10.33538157683935, 10.59476479033706, 11.17679051191512, 11.66988689264309, 11.98493498247441, 12.50341903746713, 12.97166847644236

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