Properties

Label 2-6240-1.1-c1-0-49
Degree $2$
Conductor $6240$
Sign $1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
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 − 5-s + 4.32·7-s + 9-s − 1.39·11-s + 13-s − 15-s + 5.11·17-s + 2.92·19-s + 4.32·21-s + 8.97·23-s + 25-s + 27-s − 8.36·29-s + 4·31-s − 1.39·33-s − 4.32·35-s + 3.39·37-s + 39-s − 0.601·41-s − 2.79·43-s − 45-s − 9.29·47-s + 11.6·49-s + 5.11·51-s − 2.19·53-s + 1.39·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.63·7-s + 0.333·9-s − 0.421·11-s + 0.277·13-s − 0.258·15-s + 1.24·17-s + 0.671·19-s + 0.943·21-s + 1.87·23-s + 0.200·25-s + 0.192·27-s − 1.55·29-s + 0.718·31-s − 0.243·33-s − 0.730·35-s + 0.558·37-s + 0.160·39-s − 0.0939·41-s − 0.426·43-s − 0.149·45-s − 1.35·47-s + 1.67·49-s + 0.716·51-s − 0.301·53-s + 0.188·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6240\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(49.8266\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.228861703\)
\(L(\frac12)\) \(\approx\) \(3.228861703\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 - 4.32T + 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
17 \( 1 - 5.11T + 17T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 - 8.97T + 23T^{2} \)
29 \( 1 + 8.36T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + 0.601T + 41T^{2} \)
43 \( 1 + 2.79T + 43T^{2} \)
47 \( 1 + 9.29T + 47T^{2} \)
53 \( 1 + 2.19T + 53T^{2} \)
59 \( 1 - 7.44T + 59T^{2} \)
61 \( 1 + 5.24T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 1.95T + 79T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 - 4.04T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.997560753454155666232175361418, −7.55887009451841360738584251602, −6.94229084198360791743655812930, −5.69324566495061314692373011117, −5.09325629723292996427698473080, −4.50900563267471259291440075121, −3.52185490773513273936262566005, −2.87959091015413145876618168068, −1.73450093400169879368293793299, −1.00176610861547276937527916582, 1.00176610861547276937527916582, 1.73450093400169879368293793299, 2.87959091015413145876618168068, 3.52185490773513273936262566005, 4.50900563267471259291440075121, 5.09325629723292996427698473080, 5.69324566495061314692373011117, 6.94229084198360791743655812930, 7.55887009451841360738584251602, 7.997560753454155666232175361418

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