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Properties

Label 2-2736-76.31-c1-0-46
Degree $2$
Conductor $2736$
Sign $-0.865 - 0.500i$
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  + (−2.00 − 3.47i)5-s − 1.92i·7-s − 1.38i·11-s + (0.470 + 0.271i)13-s + (−1.83 − 3.17i)17-s + (1.80 − 3.96i)19-s + (−7.04 − 4.06i)23-s + (−5.53 + 9.59i)25-s + (2.40 + 1.38i)29-s − 7.06·31-s + (−6.70 + 3.86i)35-s − 2.12i·37-s + (8.24 − 4.75i)41-s + (5.88 − 3.39i)43-s + (6.18 + 3.57i)47-s + ⋯
L(s)  = 1  + (−0.896 − 1.55i)5-s − 0.729i·7-s − 0.417i·11-s + (0.130 + 0.0753i)13-s + (−0.444 − 0.770i)17-s + (0.413 − 0.910i)19-s + (−1.46 − 0.847i)23-s + (−1.10 + 1.91i)25-s + (0.445 + 0.257i)29-s − 1.26·31-s + (−1.13 + 0.653i)35-s − 0.349i·37-s + (1.28 − 0.743i)41-s + (0.897 − 0.518i)43-s + (0.902 + 0.520i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.865 - 0.500i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7613478720\)
\(L(\frac12)\) \(\approx\) \(0.7613478720\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 3.96i)T \)
good5 \( 1 + (2.00 + 3.47i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.92iT - 7T^{2} \)
11 \( 1 + 1.38iT - 11T^{2} \)
13 \( 1 + (-0.470 - 0.271i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.04 + 4.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.40 - 1.38i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.06T + 31T^{2} \)
37 \( 1 + 2.12iT - 37T^{2} \)
41 \( 1 + (-8.24 + 4.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.88 + 3.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.18 - 3.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.171 + 0.0988i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.80 - 4.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.871 + 1.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.34 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.17 + 3.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.50 - 6.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (15.3 + 8.86i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.97 + 4.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.576619820412081161013688940715, −7.53648291468621137381048335738, −7.21182977878180459737685021485, −5.91052881699402812173066512280, −5.16163716408130982375777817079, −4.20841506543707369012317559995, −3.98199230169911727828115396320, −2.53282168173581213352506733978, −1.06194013664209475663251511265, −0.28216678945245805570632416346, 1.87702059563315465753239426091, 2.76967888627357160950112082468, 3.69035901024804603688076542773, 4.25187535474484784068884707213, 5.71391970162411269651439786114, 6.15384636209041449892200037236, 7.09683338888304467554410907606, 7.74390017974704823998911551521, 8.256940230478718447797225191534, 9.347075331250840832156961567256

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