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Properties

Label 2-399e2-1.1-c1-0-4
Degree $2$
Conductor $159201$
Sign $1$
Analytic cond. $1271.22$
Root an. cond. $35.6542$
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  − 2·4-s − 5·13-s + 4·16-s − 5·25-s + 4·31-s + 11·37-s − 13·43-s + 10·52-s + 13·61-s − 8·64-s − 16·67-s − 17·73-s + 17·79-s − 14·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s − 1.38·13-s + 16-s − 25-s + 0.718·31-s + 1.80·37-s − 1.98·43-s + 1.38·52-s + 1.66·61-s − 64-s − 1.95·67-s − 1.98·73-s + 1.91·79-s − 1.42·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159201\)    =    \(3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1271.22\)
Root analytic conductor: \(35.6542\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159201,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6792780783\)
\(L(\frac12)\) \(\approx\) \(0.6792780783\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.21878592788528, −13.01256611296387, −12.30557931253132, −11.82757566383845, −11.64594163018305, −10.82177097661674, −10.16987273015425, −9.967090694850236, −9.457052413142937, −9.108312438570539, −8.362185664299306, −8.016194634291997, −7.607364858381222, −6.960947153058113, −6.422617411883077, −5.729363188684800, −5.361555618854692, −4.684480433798374, −4.411434838184895, −3.813849826721663, −3.098202235689186, −2.594847932338125, −1.853094838402834, −1.091910029114458, −0.2685603309754153, 0.2685603309754153, 1.091910029114458, 1.853094838402834, 2.594847932338125, 3.098202235689186, 3.813849826721663, 4.411434838184895, 4.684480433798374, 5.361555618854692, 5.729363188684800, 6.422617411883077, 6.960947153058113, 7.607364858381222, 8.016194634291997, 8.362185664299306, 9.108312438570539, 9.457052413142937, 9.967090694850236, 10.16987273015425, 10.82177097661674, 11.64594163018305, 11.82757566383845, 12.30557931253132, 13.01256611296387, 13.21878592788528

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