Properties

Label 2-6348-1.1-c1-0-48
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 + 1.52·5-s + 3.74·7-s + 9-s + 1.08·11-s + 3.02·13-s + 1.52·15-s − 3.10·17-s + 4.15·19-s + 3.74·21-s − 2.67·25-s + 27-s + 2.66·29-s + 6.80·31-s + 1.08·33-s + 5.71·35-s − 6.84·37-s + 3.02·39-s + 9.95·41-s − 4.00·43-s + 1.52·45-s − 1.56·47-s + 7.04·49-s − 3.10·51-s + 2.86·53-s + 1.66·55-s + 4.15·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.682·5-s + 1.41·7-s + 0.333·9-s + 0.328·11-s + 0.839·13-s + 0.394·15-s − 0.752·17-s + 0.952·19-s + 0.817·21-s − 0.534·25-s + 0.192·27-s + 0.494·29-s + 1.22·31-s + 0.189·33-s + 0.966·35-s − 1.12·37-s + 0.484·39-s + 1.55·41-s − 0.611·43-s + 0.227·45-s − 0.228·47-s + 1.00·49-s − 0.434·51-s + 0.394·53-s + 0.223·55-s + 0.549·57-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

\(L(1)\) \(\approx\) \(3.962831745\)
\(L(\frac12)\) \(\approx\) \(3.962831745\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.52T + 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 + 3.10T + 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 - 6.80T + 31T^{2} \)
37 \( 1 + 6.84T + 37T^{2} \)
41 \( 1 - 9.95T + 41T^{2} \)
43 \( 1 + 4.00T + 43T^{2} \)
47 \( 1 + 1.56T + 47T^{2} \)
53 \( 1 - 2.86T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 - 4.63T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 - 3.21T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 3.83T + 97T^{2} \)
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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

−8.021970155559343971147410877236, −7.53978362425438653429475936887, −6.57947730510175273929079561314, −5.94875362206347756137195970537, −5.06667844263423562652400731437, −4.47508955057998780861363840157, −3.62594105460020483122440590510, −2.62894505839427235890974994476, −1.78884814285047352506616815109, −1.12944784821482962784186559573, 1.12944784821482962784186559573, 1.78884814285047352506616815109, 2.62894505839427235890974994476, 3.62594105460020483122440590510, 4.47508955057998780861363840157, 5.06667844263423562652400731437, 5.94875362206347756137195970537, 6.57947730510175273929079561314, 7.53978362425438653429475936887, 8.021970155559343971147410877236

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