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Properties

Label 2-316050-1.1-c1-0-177
Degree $2$
Conductor $316050$
Sign $-1$
Analytic cond. $2523.67$
Root an. cond. $50.2361$
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 + 8-s + 9-s − 2·11-s + 12-s − 2·13-s + 16-s − 4·17-s + 18-s + 6·19-s − 2·22-s − 6·23-s + 24-s − 2·26-s + 27-s − 10·29-s + 8·31-s + 32-s − 2·33-s − 4·34-s + 36-s − 2·37-s + 6·38-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.426·22-s − 1.25·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s − 0.685·34-s + 1/6·36-s − 0.328·37-s + 0.973·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 316050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 316050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(316050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(2523.67\)
Root analytic conductor: \(50.2361\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 316050,\ (\ :1/2),\ -1)\)

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 \( 1 \)
43 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.83473759014452, −12.63095010547228, −11.95524223804340, −11.59904488086294, −11.09494318745147, −10.74482971740212, −9.930275428520362, −9.702250335902708, −9.451472132768865, −8.530648925880206, −8.225173631980922, −7.754416084852061, −7.303574259406824, −6.810312663900519, −6.348490512721174, −5.714413618718709, −5.175029059675209, −4.918345375631385, −4.163134208456009, −3.789742739217160, −3.239167052396953, −2.662253658124470, −2.157247797249933, −1.754707322225482, −0.8362724129375440, 0, 0.8362724129375440, 1.754707322225482, 2.157247797249933, 2.662253658124470, 3.239167052396953, 3.789742739217160, 4.163134208456009, 4.918345375631385, 5.175029059675209, 5.714413618718709, 6.348490512721174, 6.810312663900519, 7.303574259406824, 7.754416084852061, 8.225173631980922, 8.530648925880206, 9.451472132768865, 9.702250335902708, 9.930275428520362, 10.74482971740212, 11.09494318745147, 11.59904488086294, 11.95524223804340, 12.63095010547228, 12.83473759014452

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