Properties

Label 2-6498-1.1-c1-0-92
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2.16·5-s + 4.36·7-s + 8-s + 2.16·10-s + 3.92·11-s − 0.954·13-s + 4.36·14-s + 16-s − 2.48·17-s + 2.16·20-s + 3.92·22-s + 7.07·23-s − 0.304·25-s − 0.954·26-s + 4.36·28-s − 9.55·29-s + 1.50·31-s + 32-s − 2.48·34-s + 9.44·35-s + 6.81·37-s + 2.16·40-s + 0.523·41-s − 9.04·43-s + 3.92·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.969·5-s + 1.64·7-s + 0.353·8-s + 0.685·10-s + 1.18·11-s − 0.264·13-s + 1.16·14-s + 0.250·16-s − 0.603·17-s + 0.484·20-s + 0.836·22-s + 1.47·23-s − 0.0608·25-s − 0.187·26-s + 0.823·28-s − 1.77·29-s + 0.269·31-s + 0.176·32-s − 0.426·34-s + 1.59·35-s + 1.12·37-s + 0.342·40-s + 0.0817·41-s − 1.37·43-s + 0.591·44-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: \(0\)
Selberg data: \((2,\ 6498,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.381327583\)
\(L(\frac12)\) \(\approx\) \(5.381327583\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.16T + 5T^{2} \)
7 \( 1 - 4.36T + 7T^{2} \)
11 \( 1 - 3.92T + 11T^{2} \)
13 \( 1 + 0.954T + 13T^{2} \)
17 \( 1 + 2.48T + 17T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 9.55T + 29T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 0.523T + 41T^{2} \)
43 \( 1 + 9.04T + 43T^{2} \)
47 \( 1 - 4.35T + 47T^{2} \)
53 \( 1 + 8.02T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 5.14T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 + 3.95T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 6.89T + 83T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−7.958093324037008001350197059945, −7.10883686499281410843450720665, −6.59881964910597318776504948549, −5.64846302955453301298798343982, −5.23233192444687546124502300054, −4.45901117850776771263448677357, −3.82822337521080219649743455820, −2.62119691730666516079554003464, −1.84689608737822213192519642954, −1.23793985303189328356049940540, 1.23793985303189328356049940540, 1.84689608737822213192519642954, 2.62119691730666516079554003464, 3.82822337521080219649743455820, 4.45901117850776771263448677357, 5.23233192444687546124502300054, 5.64846302955453301298798343982, 6.59881964910597318776504948549, 7.10883686499281410843450720665, 7.958093324037008001350197059945

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