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Properties

Label 2-163170-1.1-c1-0-5
Degree $2$
Conductor $163170$
Sign $1$
Analytic cond. $1302.91$
Root an. cond. $36.0959$
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  + 2-s + 4-s + 5-s + 8-s + 10-s − 5·11-s − 6·13-s + 16-s + 6·17-s − 3·19-s + 20-s − 5·22-s − 7·23-s + 25-s − 6·26-s + 2·29-s + 4·31-s + 32-s + 6·34-s − 37-s − 3·38-s + 40-s − 10·41-s + 7·43-s − 5·44-s − 7·46-s + 10·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.50·11-s − 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.688·19-s + 0.223·20-s − 1.06·22-s − 1.45·23-s + 1/5·25-s − 1.17·26-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.164·37-s − 0.486·38-s + 0.158·40-s − 1.56·41-s + 1.06·43-s − 0.753·44-s − 1.03·46-s + 1.45·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(163170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(1302.91\)
Root analytic conductor: \(36.0959\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 163170,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071321553\)
\(L(\frac12)\) \(\approx\) \(2.071321553\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
37 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−13.26228625494075, −12.76834443841712, −12.36161391713511, −11.83071135921399, −11.78013879219858, −10.63177233866199, −10.40563449596981, −10.00583786910106, −9.809938886454725, −8.803280421694627, −8.429359387993151, −7.683840493044964, −7.485595085083945, −7.011845688818000, −6.192026183560691, −5.726978033674991, −5.406487810440701, −4.867286488576057, −4.342910700537972, −3.780252683263362, −2.836085783914518, −2.677837028735245, −2.118390362782726, −1.361092829177052, −0.3482243339328242, 0.3482243339328242, 1.361092829177052, 2.118390362782726, 2.677837028735245, 2.836085783914518, 3.780252683263362, 4.342910700537972, 4.867286488576057, 5.406487810440701, 5.726978033674991, 6.192026183560691, 7.011845688818000, 7.485595085083945, 7.683840493044964, 8.429359387993151, 8.803280421694627, 9.809938886454725, 10.00583786910106, 10.40563449596981, 10.63177233866199, 11.78013879219858, 11.83071135921399, 12.36161391713511, 12.76834443841712, 13.26228625494075

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