Properties

Label 2-139650-1.1-c1-0-135
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 5·13-s + 16-s − 6·17-s + 18-s + 19-s + 3·23-s − 24-s − 5·26-s − 27-s − 3·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 38-s + 5·39-s + 6·41-s + 4·43-s + 3·46-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 − 0.288·12-s − 1.38·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.625·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.162·38-s + 0.800·39-s + 0.937·41-s + 0.609·43-s + 0.442·46-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: $\chi_{139650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.58673796170100, −12.99536775770272, −12.77554004799794, −12.27824497110059, −11.81382179759491, −11.21112141604470, −10.95778258995891, −10.51555783122947, −9.785455116734671, −9.344535659107025, −8.992380969142137, −8.095347359129841, −7.668844886532043, −7.040385518132079, −6.780467894012065, −6.230061537680096, −5.498752374465096, −5.167248219264484, −4.699183682535847, −4.101532857124537, −3.646978245360096, −2.739187259017893, −2.340121914937601, −1.734398310506968, −0.7778643925265246, 0, 0.7778643925265246, 1.734398310506968, 2.340121914937601, 2.739187259017893, 3.646978245360096, 4.101532857124537, 4.699183682535847, 5.167248219264484, 5.498752374465096, 6.230061537680096, 6.780467894012065, 7.040385518132079, 7.668844886532043, 8.095347359129841, 8.992380969142137, 9.344535659107025, 9.785455116734671, 10.51555783122947, 10.95778258995891, 11.21112141604470, 11.81382179759491, 12.27824497110059, 12.77554004799794, 12.99536775770272, 13.58673796170100

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