Properties

Label 2-2496-1.1-c3-0-98
Degree $2$
Conductor $2496$
Sign $1$
Analytic cond. $147.268$
Root an. cond. $12.1354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 22.0·5-s − 0.570·7-s + 9·9-s + 14.5·11-s + 13·13-s + 66.2·15-s + 62.8·17-s + 106.·19-s − 1.71·21-s + 175.·23-s + 362.·25-s + 27·27-s − 66.4·29-s + 53.0·31-s + 43.5·33-s − 12.6·35-s − 61.7·37-s + 39·39-s − 244.·41-s − 64.2·43-s + 198.·45-s − 51.2·47-s − 342.·49-s + 188.·51-s − 495.·53-s + 320.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.97·5-s − 0.0308·7-s + 0.333·9-s + 0.397·11-s + 0.277·13-s + 1.14·15-s + 0.896·17-s + 1.29·19-s − 0.0177·21-s + 1.59·23-s + 2.90·25-s + 0.192·27-s − 0.425·29-s + 0.307·31-s + 0.229·33-s − 0.0608·35-s − 0.274·37-s + 0.160·39-s − 0.932·41-s − 0.227·43-s + 0.658·45-s − 0.159·47-s − 0.999·49-s + 0.517·51-s − 1.28·53-s + 0.786·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(147.268\)
Root analytic conductor: \(12.1354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.451488592\)
\(L(\frac12)\) \(\approx\) \(5.451488592\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
13 \( 1 - 13T \)
good5 \( 1 - 22.0T + 125T^{2} \)
7 \( 1 + 0.570T + 343T^{2} \)
11 \( 1 - 14.5T + 1.33e3T^{2} \)
17 \( 1 - 62.8T + 4.91e3T^{2} \)
19 \( 1 - 106.T + 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 + 66.4T + 2.43e4T^{2} \)
31 \( 1 - 53.0T + 2.97e4T^{2} \)
37 \( 1 + 61.7T + 5.06e4T^{2} \)
41 \( 1 + 244.T + 6.89e4T^{2} \)
43 \( 1 + 64.2T + 7.95e4T^{2} \)
47 \( 1 + 51.2T + 1.03e5T^{2} \)
53 \( 1 + 495.T + 1.48e5T^{2} \)
59 \( 1 + 322.T + 2.05e5T^{2} \)
61 \( 1 + 499.T + 2.26e5T^{2} \)
67 \( 1 - 145.T + 3.00e5T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 + 592.T + 3.89e5T^{2} \)
79 \( 1 - 124.T + 4.93e5T^{2} \)
83 \( 1 - 675.T + 5.71e5T^{2} \)
89 \( 1 + 411.T + 7.04e5T^{2} \)
97 \( 1 + 1.63e3T + 9.12e5T^{2} \)
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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

−8.824102380878384696876027808153, −7.85137803728025732816797145986, −6.89786408900404409487932934957, −6.30439048462185351231838246011, −5.38315572810360131393032772728, −4.90760038586283936388602550169, −3.37484104517929386813655253404, −2.83394862210132988753720353654, −1.65735120965258510740403091322, −1.14679825277975694578224031185, 1.14679825277975694578224031185, 1.65735120965258510740403091322, 2.83394862210132988753720353654, 3.37484104517929386813655253404, 4.90760038586283936388602550169, 5.38315572810360131393032772728, 6.30439048462185351231838246011, 6.89786408900404409487932934957, 7.85137803728025732816797145986, 8.824102380878384696876027808153

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