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Properties

Label 2-308550-1.1-c1-0-138
Degree $2$
Conductor $308550$
Sign $-1$
Analytic cond. $2463.78$
Root an. cond. $49.6365$
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 − 4·7-s + 8-s + 9-s − 12-s − 4·13-s − 4·14-s + 16-s − 17-s + 18-s + 8·19-s + 4·21-s − 24-s − 4·26-s − 27-s − 4·28-s + 10·31-s + 32-s − 34-s + 36-s − 8·37-s + 8·38-s + 4·39-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.872·21-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.755·28-s + 1.79·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s − 1.31·37-s + 1.29·38-s + 0.640·39-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2463.78\)
Root analytic conductor: \(49.6365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 308550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.93071252581131, −12.30916941743384, −12.05316869839254, −11.74919490309803, −11.24055566870632, −10.52978898659547, −10.12787136175184, −9.800765377578144, −9.452709232106275, −8.861856448028189, −8.138368878281399, −7.576060270053979, −7.121586062433245, −6.749185249387471, −6.329736716642619, −5.812697641438594, −5.287496315542990, −4.922921610923143, −4.355805070315064, −3.750566820029478, −3.110866680905050, −2.912832870834590, −2.239745571838488, −1.397653150690738, −0.7054223720528394, 0, 0.7054223720528394, 1.397653150690738, 2.239745571838488, 2.912832870834590, 3.110866680905050, 3.750566820029478, 4.355805070315064, 4.922921610923143, 5.287496315542990, 5.812697641438594, 6.329736716642619, 6.749185249387471, 7.121586062433245, 7.576060270053979, 8.138368878281399, 8.861856448028189, 9.452709232106275, 9.800765377578144, 10.12787136175184, 10.52978898659547, 11.24055566870632, 11.74919490309803, 12.05316869839254, 12.30916941743384, 12.93071252581131

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