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Properties

Label 2-2736-57.56-c1-0-17
Degree $2$
Conductor $2736$
Sign $0.881 - 0.471i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0959i·5-s + 3.22·7-s + 0.634i·11-s − 1.00i·13-s + 3.46i·17-s + (−3.89 − 1.95i)19-s + 6.51i·23-s + 4.99·25-s + 6.76·29-s − 9.20i·31-s + 0.309i·35-s + 10.9i·37-s + 10.2·41-s + 0.906·43-s + 6.82i·47-s + ⋯
L(s)  = 1  + 0.0429i·5-s + 1.22·7-s + 0.191i·11-s − 0.279i·13-s + 0.839i·17-s + (−0.894 − 0.447i)19-s + 1.35i·23-s + 0.998·25-s + 1.25·29-s − 1.65i·31-s + 0.0523i·35-s + 1.80i·37-s + 1.60·41-s + 0.138·43-s + 0.996i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.122409701\)
\(L(\frac12)\) \(\approx\) \(2.122409701\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (3.89 + 1.95i)T \)
good5 \( 1 - 0.0959iT - 5T^{2} \)
7 \( 1 - 3.22T + 7T^{2} \)
11 \( 1 - 0.634iT - 11T^{2} \)
13 \( 1 + 1.00iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
23 \( 1 - 6.51iT - 23T^{2} \)
29 \( 1 - 6.76T + 29T^{2} \)
31 \( 1 + 9.20iT - 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 0.906T + 43T^{2} \)
47 \( 1 - 6.82iT - 47T^{2} \)
53 \( 1 + 4.18T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 7.41T + 61T^{2} \)
67 \( 1 + 4.28iT - 67T^{2} \)
71 \( 1 - 5.42T + 71T^{2} \)
73 \( 1 + 1.75T + 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + 3.62iT - 83T^{2} \)
89 \( 1 + 3.30T + 89T^{2} \)
97 \( 1 - 6.30iT - 97T^{2} \)
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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

−8.776115701116605145927930033516, −8.032462964429097664631135220002, −7.62117655667923789080129987842, −6.51791676294884579089594460307, −5.85610124166144373301429501752, −4.78253950189916976760102351916, −4.38375722858445606059643867806, −3.15642228375035551486873226039, −2.10590720767362227203304312834, −1.10639823159059708129288981131, 0.826023632504338100482834674483, 2.02187902721092785792854360929, 2.92689915201261908055333115147, 4.24224001418013745445155558727, 4.75077801034516104346771280024, 5.57508044268515153911605541401, 6.59167006819161500047273321843, 7.20812701754140098404074352214, 8.224879471350593214993443137724, 8.591995749380048987541602069600

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