Properties

Label 2-6348-1.1-c1-0-32
Degree $2$
Conductor $6348$
Sign $1$
Analytic cond. $50.6890$
Root an. cond. $7.11962$
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  + 3-s − 2.68·5-s + 0.168·7-s + 9-s + 5.32·11-s + 3.84·13-s − 2.68·15-s + 6.65·17-s + 6.89·19-s + 0.168·21-s + 2.22·25-s + 27-s − 0.569·29-s − 6.95·31-s + 5.32·33-s − 0.452·35-s − 8.85·37-s + 3.84·39-s + 4.86·41-s + 6.75·43-s − 2.68·45-s − 2.26·47-s − 6.97·49-s + 6.65·51-s − 4.87·53-s − 14.3·55-s + 6.89·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.20·5-s + 0.0636·7-s + 0.333·9-s + 1.60·11-s + 1.06·13-s − 0.693·15-s + 1.61·17-s + 1.58·19-s + 0.0367·21-s + 0.444·25-s + 0.192·27-s − 0.105·29-s − 1.24·31-s + 0.926·33-s − 0.0765·35-s − 1.45·37-s + 0.615·39-s + 0.760·41-s + 1.03·43-s − 0.400·45-s − 0.330·47-s − 0.995·49-s + 0.932·51-s − 0.669·53-s − 1.92·55-s + 0.912·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6348\)    =    \(2^{2} \cdot 3 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(50.6890\)
Root analytic conductor: \(7.11962\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6348,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.669632603\)
\(L(\frac12)\) \(\approx\) \(2.669632603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 \)
good5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 - 0.168T + 7T^{2} \)
11 \( 1 - 5.32T + 11T^{2} \)
13 \( 1 - 3.84T + 13T^{2} \)
17 \( 1 - 6.65T + 17T^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
29 \( 1 + 0.569T + 29T^{2} \)
31 \( 1 + 6.95T + 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 - 4.86T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 + 2.26T + 47T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 - 2.28T + 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 - 1.61T + 67T^{2} \)
71 \( 1 + 9.49T + 71T^{2} \)
73 \( 1 - 3.83T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 - 8.96T + 83T^{2} \)
89 \( 1 + 6.28T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−7.963491369011610397559176303094, −7.45746585673372788802758114284, −6.85130923331045934831847373491, −5.90905293970855887840837611012, −5.15348072636107860690931630199, −4.01473515985175138478749347751, −3.63527919272159907503946294838, −3.17296635549267356678564578656, −1.61429094421111388759665647659, −0.918588765620667609138690879049, 0.918588765620667609138690879049, 1.61429094421111388759665647659, 3.17296635549267356678564578656, 3.63527919272159907503946294838, 4.01473515985175138478749347751, 5.15348072636107860690931630199, 5.90905293970855887840837611012, 6.85130923331045934831847373491, 7.45746585673372788802758114284, 7.963491369011610397559176303094

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