Properties

Label 2-63175-1.1-c1-0-10
Degree $2$
Conductor $63175$
Sign $1$
Analytic cond. $504.454$
Root an. cond. $22.4600$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 7-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s + 4·16-s − 3·17-s − 21-s + 6·23-s − 5·27-s + 2·28-s − 3·29-s + 4·31-s − 3·33-s + 4·36-s + 2·37-s + 5·39-s + 12·41-s + 10·43-s + 6·44-s − 9·47-s + 4·48-s + 49-s − 3·51-s − 10·52-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s + 16-s − 0.727·17-s − 0.218·21-s + 1.25·23-s − 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.718·31-s − 0.522·33-s + 2/3·36-s + 0.328·37-s + 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.904·44-s − 1.31·47-s + 0.577·48-s + 1/7·49-s − 0.420·51-s − 1.38·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63175\)    =    \(5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(504.454\)
Root analytic conductor: \(22.4600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{63175} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 63175,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.809862154\)
\(L(\frac12)\) \(\approx\) \(1.809862154\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.32209000634248, −13.53910094576619, −13.34742517398724, −12.96845871997650, −12.53847673007899, −11.59632434684325, −11.18359859625835, −10.69651885373280, −10.14953721032221, −9.424434418439977, −9.086083923609679, −8.617714978969786, −8.272708934397661, −7.624449965279317, −7.115465414127886, −6.095063466531719, −5.928652731494114, −5.218362481680253, −4.591824772789197, −3.948168418175215, −3.458943721185237, −2.787242575934683, −2.306108168015885, −1.137233769903359, −0.5031378884281167, 0.5031378884281167, 1.137233769903359, 2.306108168015885, 2.787242575934683, 3.458943721185237, 3.948168418175215, 4.591824772789197, 5.218362481680253, 5.928652731494114, 6.095063466531719, 7.115465414127886, 7.624449965279317, 8.272708934397661, 8.617714978969786, 9.086083923609679, 9.424434418439977, 10.14953721032221, 10.69651885373280, 11.18359859625835, 11.59632434684325, 12.53847673007899, 12.96845871997650, 13.34742517398724, 13.53910094576619, 14.32209000634248

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