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Properties

Label 2-2736-76.31-c1-0-17
Degree $2$
Conductor $2736$
Sign $-0.595 - 0.802i$
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  + (0.912 + 1.58i)5-s + 4.99i·7-s + 3.82i·11-s + (1.00 + 0.581i)13-s + (3.73 + 6.47i)17-s + (3.22 − 2.92i)19-s + (−2.24 − 1.29i)23-s + (0.833 − 1.44i)25-s + (6.63 + 3.82i)29-s + 0.158·31-s + (−7.89 + 4.55i)35-s − 8.25i·37-s + (1.07 − 0.617i)41-s + (−1.90 + 1.09i)43-s + (−0.0858 − 0.0495i)47-s + ⋯
L(s)  = 1  + (0.408 + 0.707i)5-s + 1.88i·7-s + 1.15i·11-s + (0.279 + 0.161i)13-s + (0.906 + 1.56i)17-s + (0.740 − 0.671i)19-s + (−0.468 − 0.270i)23-s + (0.166 − 0.288i)25-s + (1.23 + 0.711i)29-s + 0.0285·31-s + (−1.33 + 0.770i)35-s − 1.35i·37-s + (0.167 − 0.0965i)41-s + (−0.290 + 0.167i)43-s + (−0.0125 − 0.00722i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.595 - 0.802i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.595 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038551135\)
\(L(\frac12)\) \(\approx\) \(2.038551135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 + 2.92i)T \)
good5 \( 1 + (-0.912 - 1.58i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.99iT - 7T^{2} \)
11 \( 1 - 3.82iT - 11T^{2} \)
13 \( 1 + (-1.00 - 0.581i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.73 - 6.47i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.24 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.63 - 3.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.158T + 31T^{2} \)
37 \( 1 + 8.25iT - 37T^{2} \)
41 \( 1 + (-1.07 + 0.617i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 - 1.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0858 + 0.0495i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 + 1.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.14 - 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.64 + 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.97 - 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.91 - 3.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.32 + 4.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.74 + 8.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.83iT - 83T^{2} \)
89 \( 1 + (11.9 + 6.87i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.50 + 4.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.961695373661299960773046848840, −8.527414896675656961633747054413, −7.56813394689215891847967347575, −6.66892411635686996249215111466, −6.00079899896640399820702367601, −5.40860594071389710066986062105, −4.44781987605161734389934946078, −3.21954720883841659214881224679, −2.44737189503110599509382383145, −1.67568525033909616587661307603, 0.74905100848128333388540639173, 1.23557446891345144969477316740, 3.02777641780504543386511382475, 3.67995828133241433478419662121, 4.68508538202235057192469912221, 5.36283602546752695452054855902, 6.26959312477467763401193249121, 7.12095527132357879538947563611, 7.86270377173232450619849264714, 8.389239367339897370608438488073

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