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Properties

Label 2-139650-1.1-c1-0-250
Degree $2$
Conductor $139650$
Sign $1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 5·11-s + 12-s − 5·13-s + 16-s − 2·17-s + 18-s + 19-s − 5·22-s − 9·23-s + 24-s − 5·26-s + 27-s − 6·29-s − 9·31-s + 32-s − 5·33-s − 2·34-s + 36-s − 10·37-s + 38-s − 5·39-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 − 1.50·11-s + 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 1.06·22-s − 1.87·23-s + 0.204·24-s − 0.980·26-s + 0.192·27-s − 1.11·29-s − 1.61·31-s + 0.176·32-s − 0.870·33-s − 0.342·34-s + 1/6·36-s − 1.64·37-s + 0.162·38-s − 0.800·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

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 \( 1 \)
19 \( 1 - T \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.78099112326047, −13.54702693565985, −12.92490580066654, −12.51704286660936, −12.20915771205877, −11.60064426263773, −10.99634288478908, −10.50752430743623, −10.14610951981548, −9.482935909786347, −9.267735330658930, −8.262725316470935, −7.992981698727249, −7.596148453921426, −7.005550249090542, −6.592358768427591, −5.770419181602842, −5.226668371454203, −5.032552153312314, −4.318444148174169, −3.602881171497254, −3.307141482707620, −2.463803464307008, −2.080710636772729, −1.667207603793225, 0, 0, 1.667207603793225, 2.080710636772729, 2.463803464307008, 3.307141482707620, 3.602881171497254, 4.318444148174169, 5.032552153312314, 5.226668371454203, 5.770419181602842, 6.592358768427591, 7.005550249090542, 7.596148453921426, 7.992981698727249, 8.262725316470935, 9.267735330658930, 9.482935909786347, 10.14610951981548, 10.50752430743623, 10.99634288478908, 11.60064426263773, 12.20915771205877, 12.51704286660936, 12.92490580066654, 13.54702693565985, 13.78099112326047

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