Properties

Label 2-2736-19.13-c0-0-0
Degree $2$
Conductor $2736$
Sign $0.939 + 0.342i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 1.62i)7-s + (0.673 + 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)25-s + (1.11 + 0.642i)31-s − 1.28i·37-s + (−0.266 − 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (−0.439 + 0.524i)67-s + (−0.326 − 0.118i)73-s + (0.233 − 0.642i)79-s + (3.64 + 0.642i)91-s + (−1.11 − 1.32i)97-s + (−1.70 + 0.984i)103-s + ⋯
L(s)  = 1  + (0.939 − 1.62i)7-s + (0.673 + 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)25-s + (1.11 + 0.642i)31-s − 1.28i·37-s + (−0.266 − 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (−0.439 + 0.524i)67-s + (−0.326 − 0.118i)73-s + (0.233 − 0.642i)79-s + (3.64 + 0.642i)91-s + (−1.11 − 1.32i)97-s + (−1.70 + 0.984i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419796788\)
\(L(\frac12)\) \(\approx\) \(1.419796788\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.28iT - T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
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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

−8.755677205042851612416359342992, −8.326435740993246523863298795078, −7.27341200647528206133582853837, −6.90257710591046892379693291440, −6.01391715720720811382677776282, −4.78303606592705515665356501300, −4.24668603952314352125614261726, −3.60817876515959616251396104805, −2.03357108038411536476503018909, −1.17747433708757185479128214790, 1.27074827134321347631009450844, 2.58612481982095659005971066799, 3.08927906755982887035253645996, 4.57819724793505743885516438634, 5.19219817562293117150937858841, 5.89192992367158226041859140280, 6.60343589740727541380990051893, 7.925001884022442366157726085755, 8.232115303200598623719385776786, 8.878181913091904374681772333832

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