Properties

Label 2-6348-1.1-c1-0-77
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.04·5-s + 1.46·7-s + 9-s − 2.04·11-s − 6.89·13-s + 1.04·15-s − 7.77·17-s + 6.37·19-s + 1.46·21-s − 3.90·25-s + 27-s + 4.14·29-s + 0.909·31-s − 2.04·33-s + 1.53·35-s + 4.07·37-s − 6.89·39-s + 4.89·41-s − 3.50·43-s + 1.04·45-s − 1.92·47-s − 4.85·49-s − 7.77·51-s − 7.82·53-s − 2.14·55-s + 6.37·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.468·5-s + 0.552·7-s + 0.333·9-s − 0.617·11-s − 1.91·13-s + 0.270·15-s − 1.88·17-s + 1.46·19-s + 0.319·21-s − 0.780·25-s + 0.192·27-s + 0.770·29-s + 0.163·31-s − 0.356·33-s + 0.258·35-s + 0.670·37-s − 1.10·39-s + 0.764·41-s − 0.534·43-s + 0.156·45-s − 0.281·47-s − 0.694·49-s − 1.08·51-s − 1.07·53-s − 0.289·55-s + 0.844·57-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: \(1\)
Selberg data: \((2,\ 6348,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
23 \( 1 \)
good5 \( 1 - 1.04T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 + 2.04T + 11T^{2} \)
13 \( 1 + 6.89T + 13T^{2} \)
17 \( 1 + 7.77T + 17T^{2} \)
19 \( 1 - 6.37T + 19T^{2} \)
29 \( 1 - 4.14T + 29T^{2} \)
31 \( 1 - 0.909T + 31T^{2} \)
37 \( 1 - 4.07T + 37T^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 + 3.50T + 43T^{2} \)
47 \( 1 + 1.92T + 47T^{2} \)
53 \( 1 + 7.82T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 + 7.53T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 0.197T + 71T^{2} \)
73 \( 1 + 7.96T + 73T^{2} \)
79 \( 1 + 4.62T + 79T^{2} \)
83 \( 1 - 8.28T + 83T^{2} \)
89 \( 1 + 0.275T + 89T^{2} \)
97 \( 1 + 7.35T + 97T^{2} \)
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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

−7.71214918718823883568050248058, −7.13075249929824912666213514764, −6.37013122401533790140969658246, −5.33746463113187742987173644371, −4.82149967159521450054225825227, −4.17414846076972141810006949441, −2.83676135520376319895738740604, −2.46925220966909041090201321314, −1.54845082355620012018223062559, 0, 1.54845082355620012018223062559, 2.46925220966909041090201321314, 2.83676135520376319895738740604, 4.17414846076972141810006949441, 4.82149967159521450054225825227, 5.33746463113187742987173644371, 6.37013122401533790140969658246, 7.13075249929824912666213514764, 7.71214918718823883568050248058

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