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Properties

Label 2-303450-1.1-c1-0-103
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 − 14-s + 16-s + 18-s + 8·19-s + 21-s − 4·22-s − 24-s − 27-s − 28-s − 6·29-s + 32-s + 4·33-s + 36-s − 2·37-s + 8·38-s − 2·41-s + 42-s − 6·43-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 − 0.267·14-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.218·21-s − 0.852·22-s − 0.204·24-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.696·33-s + 1/6·36-s − 0.328·37-s + 1.29·38-s − 0.312·41-s + 0.154·42-s − 0.914·43-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−12.98097548228812, −12.52396929891301, −11.93431278620642, −11.64408712709477, −11.14017770136844, −10.78262016054175, −10.16204672264781, −9.817362360492204, −9.446054666016174, −8.752251405050339, −8.087003341204459, −7.631191990850397, −7.337952213712349, −6.746173296096119, −6.224706771122386, −5.761040990240020, −5.250449253796622, −4.957962057793720, −4.508475990071042, −3.605390847686819, −3.276018420106234, −2.903935181132635, −2.000567954424323, −1.596848619373328, −0.7030135716694316, 0, 0.7030135716694316, 1.596848619373328, 2.000567954424323, 2.903935181132635, 3.276018420106234, 3.605390847686819, 4.508475990071042, 4.957962057793720, 5.250449253796622, 5.761040990240020, 6.224706771122386, 6.746173296096119, 7.337952213712349, 7.631191990850397, 8.087003341204459, 8.752251405050339, 9.446054666016174, 9.817362360492204, 10.16204672264781, 10.78262016054175, 11.14017770136844, 11.64408712709477, 11.93431278620642, 12.52396929891301, 12.98097548228812

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