Properties

Label 2-102-1.1-c3-0-2
Degree $2$
Conductor $102$
Sign $1$
Analytic cond. $6.01819$
Root an. cond. $2.45320$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s + 12·7-s + 8·8-s + 9·9-s + 10·10-s + 37·11-s − 12·12-s + 19·13-s + 24·14-s − 15·15-s + 16·16-s + 17·17-s + 18·18-s + 37·19-s + 20·20-s − 36·21-s + 74·22-s − 3·23-s − 24·24-s − 100·25-s + 38·26-s − 27·27-s + 48·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.647·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.01·11-s − 0.288·12-s + 0.405·13-s + 0.458·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.446·19-s + 0.223·20-s − 0.374·21-s + 0.717·22-s − 0.0271·23-s − 0.204·24-s − 4/5·25-s + 0.286·26-s − 0.192·27-s + 0.323·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $1$
Analytic conductor: \(6.01819\)
Root analytic conductor: \(2.45320\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.330635747\)
\(L(\frac12)\) \(\approx\) \(2.330635747\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
17 \( 1 - p T \)
good5 \( 1 - p T + p^{3} T^{2} \)
7 \( 1 - 12 T + p^{3} T^{2} \)
11 \( 1 - 37 T + p^{3} T^{2} \)
13 \( 1 - 19 T + p^{3} T^{2} \)
19 \( 1 - 37 T + p^{3} T^{2} \)
23 \( 1 + 3 T + p^{3} T^{2} \)
29 \( 1 + 86 T + p^{3} T^{2} \)
31 \( 1 + 142 T + p^{3} T^{2} \)
37 \( 1 + 8 p T + p^{3} T^{2} \)
41 \( 1 + 121 T + p^{3} T^{2} \)
43 \( 1 - 3 T + p^{3} T^{2} \)
47 \( 1 - 402 T + p^{3} T^{2} \)
53 \( 1 - 174 T + p^{3} T^{2} \)
59 \( 1 - 270 T + p^{3} T^{2} \)
61 \( 1 + 520 T + p^{3} T^{2} \)
67 \( 1 + 780 T + p^{3} T^{2} \)
71 \( 1 - 84 T + p^{3} T^{2} \)
73 \( 1 + 302 T + p^{3} T^{2} \)
79 \( 1 - 178 T + p^{3} T^{2} \)
83 \( 1 - 698 T + p^{3} T^{2} \)
89 \( 1 - 1512 T + p^{3} T^{2} \)
97 \( 1 + 500 T + p^{3} T^{2} \)
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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

−13.44092654516132302689916159958, −12.18434491403111496489834604027, −11.46923469239353842591973267823, −10.41208178909743828330693651066, −9.071804812449913545579322418352, −7.47169879447251191436078965196, −6.21924868425965038901060308368, −5.21227854404398461206954567051, −3.80875045662764260703873870255, −1.62707210108612188764142658187, 1.62707210108612188764142658187, 3.80875045662764260703873870255, 5.21227854404398461206954567051, 6.21924868425965038901060308368, 7.47169879447251191436078965196, 9.071804812449913545579322418352, 10.41208178909743828330693651066, 11.46923469239353842591973267823, 12.18434491403111496489834604027, 13.44092654516132302689916159958

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