Properties

Label 2-303450-1.1-c1-0-127
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 + 13-s + 14-s + 16-s − 18-s − 4·19-s − 21-s + 3·22-s + 6·23-s − 24-s − 26-s + 27-s − 28-s − 8·29-s + 7·31-s − 32-s − 3·33-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 − 0.904·11-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 + 0.639·22-s + 1.25·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.48·29-s + 1.25·31-s − 0.176·32-s − 0.522·33-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: \(1\)
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 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−12.95388698109769, −12.50721638310061, −12.01144266866643, −11.31985685708892, −11.02524706104596, −10.52485996347340, −10.14344591973864, −9.673973233091389, −9.005914878806618, −8.969985466946001, −8.296298786315874, −7.874476932724701, −7.358178782819505, −7.102741763324520, −6.326161785806057, −6.026407569991417, −5.399230413327914, −4.830683490597641, −4.117191580284923, −3.781892657367772, −2.968750119306308, −2.519566919748145, −2.279526947942561, −1.306211364549507, −0.8038914019793767, 0, 0.8038914019793767, 1.306211364549507, 2.279526947942561, 2.519566919748145, 2.968750119306308, 3.781892657367772, 4.117191580284923, 4.830683490597641, 5.399230413327914, 6.026407569991417, 6.326161785806057, 7.102741763324520, 7.358178782819505, 7.874476932724701, 8.296298786315874, 8.969985466946001, 9.005914878806618, 9.673973233091389, 10.14344591973864, 10.52485996347340, 11.02524706104596, 11.31985685708892, 12.01144266866643, 12.50721638310061, 12.95388698109769

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