Properties

Label 2-12138-1.1-c1-0-22
Degree $2$
Conductor $12138$
Sign $-1$
Analytic cond. $96.9224$
Root an. cond. $9.84491$
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 − 3·5-s + 6-s − 7-s + 8-s + 9-s − 3·10-s − 5·11-s + 12-s − 14-s − 3·15-s + 16-s + 18-s + 6·19-s − 3·20-s − 21-s − 5·22-s + 2·23-s + 24-s + 4·25-s + 27-s − 28-s + 9·29-s − 3·30-s + 3·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.50·11-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.670·20-s − 0.218·21-s − 1.06·22-s + 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.547·30-s + 0.538·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12138\)    =    \(2 \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(96.9224\)
Root analytic conductor: \(9.84491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12138} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12138,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−16.19495566706007, −15.91193673818095, −15.40387980834494, −15.23849008271074, −14.32056817301614, −13.60776658106121, −13.51505965353980, −12.55465488441039, −12.17141857537982, −11.72700963141058, −10.90522488694232, −10.39271047566692, −9.827521323326749, −8.908059373833136, −8.177535428266084, −7.862952837630769, −7.171451098756134, −6.710785662223461, −5.641986465241105, −4.989144416539782, −4.452426041944807, −3.568618824594263, −3.061754991577573, −2.570736027190322, −1.219366133849524, 0, 1.219366133849524, 2.570736027190322, 3.061754991577573, 3.568618824594263, 4.452426041944807, 4.989144416539782, 5.641986465241105, 6.710785662223461, 7.171451098756134, 7.862952837630769, 8.177535428266084, 8.908059373833136, 9.827521323326749, 10.39271047566692, 10.90522488694232, 11.72700963141058, 12.17141857537982, 12.55465488441039, 13.51505965353980, 13.60776658106121, 14.32056817301614, 15.23849008271074, 15.40387980834494, 15.91193673818095, 16.19495566706007

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