Properties

Label 2-6498-1.1-c1-0-136
Degree $2$
Conductor $6498$
Sign $-1$
Analytic cond. $51.8867$
Root an. cond. $7.20324$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2.34·5-s − 1.28·7-s + 8-s + 2.34·10-s − 5.75·11-s − 0.304·13-s − 1.28·14-s + 16-s − 4.18·17-s + 2.34·20-s − 5.75·22-s + 6.47·23-s + 0.497·25-s − 0.304·26-s − 1.28·28-s − 3.12·29-s − 6.44·31-s + 32-s − 4.18·34-s − 3.01·35-s + 3.97·37-s + 2.34·40-s − 5.01·41-s − 0.989·43-s − 5.75·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.04·5-s − 0.485·7-s + 0.353·8-s + 0.741·10-s − 1.73·11-s − 0.0843·13-s − 0.343·14-s + 0.250·16-s − 1.01·17-s + 0.524·20-s − 1.22·22-s + 1.35·23-s + 0.0994·25-s − 0.0596·26-s − 0.242·28-s − 0.580·29-s − 1.15·31-s + 0.176·32-s − 0.717·34-s − 0.508·35-s + 0.654·37-s + 0.370·40-s − 0.783·41-s − 0.150·43-s − 0.867·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6498\)    =    \(2 \cdot 3^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(51.8867\)
Root analytic conductor: \(7.20324\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6498,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + 0.304T + 13T^{2} \)
17 \( 1 + 4.18T + 17T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + 3.12T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 + 0.989T + 43T^{2} \)
47 \( 1 - 4.39T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 - 3.31T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 4.38T + 67T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 + 2.26T + 73T^{2} \)
79 \( 1 + 8.57T + 79T^{2} \)
83 \( 1 - 9.76T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−7.43818020376071750942532793548, −6.87541629228144831782447653635, −6.09482418143873260348988176613, −5.42748147601726906331582804849, −5.03039145691331240965360041576, −4.06581922281404590057284784195, −2.99071172836294244258984442077, −2.51195860906334722378037569372, −1.63896316358874672567725334531, 0, 1.63896316358874672567725334531, 2.51195860906334722378037569372, 2.99071172836294244258984442077, 4.06581922281404590057284784195, 5.03039145691331240965360041576, 5.42748147601726906331582804849, 6.09482418143873260348988176613, 6.87541629228144831782447653635, 7.43818020376071750942532793548

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