Properties

Label 2-2736-19.18-c2-0-31
Degree $2$
Conductor $2736$
Sign $-0.386 - 0.922i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·5-s − 4.59·7-s + 13.7·11-s + 22.2i·13-s − 5.43·17-s + (7.35 + 17.5i)19-s + 24.9·23-s − 19.6·25-s − 34.7i·29-s + 41.2i·31-s − 10.6·35-s − 20.5i·37-s + 27.8i·41-s + 10.8·43-s − 33.8·47-s + ⋯
L(s)  = 1  + 0.461·5-s − 0.656·7-s + 1.25·11-s + 1.71i·13-s − 0.319·17-s + (0.386 + 0.922i)19-s + 1.08·23-s − 0.787·25-s − 1.19i·29-s + 1.33i·31-s − 0.302·35-s − 0.556i·37-s + 0.680i·41-s + 0.252·43-s − 0.720·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.386 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.631115987\)
\(L(\frac12)\) \(\approx\) \(1.631115987\)
\(L(2)\) 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 \( 1 \)
19 \( 1 + (-7.35 - 17.5i)T \)
good5 \( 1 - 2.30T + 25T^{2} \)
7 \( 1 + 4.59T + 49T^{2} \)
11 \( 1 - 13.7T + 121T^{2} \)
13 \( 1 - 22.2iT - 169T^{2} \)
17 \( 1 + 5.43T + 289T^{2} \)
23 \( 1 - 24.9T + 529T^{2} \)
29 \( 1 + 34.7iT - 841T^{2} \)
31 \( 1 - 41.2iT - 961T^{2} \)
37 \( 1 + 20.5iT - 1.36e3T^{2} \)
41 \( 1 - 27.8iT - 1.68e3T^{2} \)
43 \( 1 - 10.8T + 1.84e3T^{2} \)
47 \( 1 + 33.8T + 2.20e3T^{2} \)
53 \( 1 + 29.2iT - 2.80e3T^{2} \)
59 \( 1 + 65.8iT - 3.48e3T^{2} \)
61 \( 1 + 107.T + 3.72e3T^{2} \)
67 \( 1 - 1.78iT - 4.48e3T^{2} \)
71 \( 1 - 95.8iT - 5.04e3T^{2} \)
73 \( 1 - 65.4T + 5.32e3T^{2} \)
79 \( 1 + 37.2iT - 6.24e3T^{2} \)
83 \( 1 + 32.6T + 6.88e3T^{2} \)
89 \( 1 - 74.3iT - 7.92e3T^{2} \)
97 \( 1 + 128. iT - 9.40e3T^{2} \)
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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

−9.185523687706053890258035062418, −8.224985258067234488779379983110, −7.15674751373323756503649282264, −6.50817405947655591212637790876, −6.09791372677776322826413316821, −4.92891322146842035934634992113, −4.06943983917399422964858905872, −3.34712281582637704333108050097, −2.07860768429667855068104498073, −1.28017745545147991243189370322, 0.38374867134556819489130797406, 1.45537485103138897713399875789, 2.80036731183067688368693806443, 3.38945582260778346437203333837, 4.48504043035604581268258517928, 5.42857207182010596011706314027, 6.10366844381233144578784679189, 6.84610259027599516094797969476, 7.56899270954060002803258681730, 8.521660717295104277706213045556

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