Properties

Label 2-303450-1.1-c1-0-113
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 + 4·11-s − 12-s + 3·13-s + 14-s + 16-s − 18-s − 2·19-s + 21-s − 4·22-s + 9·23-s + 24-s − 3·26-s − 27-s − 28-s − 8·29-s + 10·31-s − 32-s − 4·33-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 + 1.20·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.218·21-s − 0.852·22-s + 1.87·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.48·29-s + 1.79·31-s − 0.176·32-s − 0.696·33-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: $\chi_{303450} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−12.87396390764894, −12.42400857774301, −11.68702109530578, −11.56602149801926, −10.99613316734525, −10.80052710291324, −10.10206733370979, −9.615745522452400, −9.231117200109752, −8.958295407922777, −8.281694142700992, −7.855019914830697, −7.312922072431671, −6.695750821972321, −6.336360038623000, −6.178564964599589, −5.472416523429485, −4.723500953643751, −4.401942768764561, −3.727462425441502, −3.066182941783648, −2.789964733352083, −1.582012395209424, −1.482625410758411, −0.7634801971399202, 0, 0.7634801971399202, 1.482625410758411, 1.582012395209424, 2.789964733352083, 3.066182941783648, 3.727462425441502, 4.401942768764561, 4.723500953643751, 5.472416523429485, 6.178564964599589, 6.336360038623000, 6.695750821972321, 7.312922072431671, 7.855019914830697, 8.281694142700992, 8.958295407922777, 9.231117200109752, 9.615745522452400, 10.10206733370979, 10.80052710291324, 10.99613316734525, 11.56602149801926, 11.68702109530578, 12.42400857774301, 12.87396390764894

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