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Properties

Label 2-2736-57.8-c1-0-39
Degree $2$
Conductor $2736$
Sign $-0.998 - 0.0586i$
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  + (0.0242 + 0.0140i)5-s + 1.95·7-s − 2.26i·11-s + (−1.04 + 0.605i)13-s + (−5.68 − 3.27i)17-s + (−3.90 + 1.93i)19-s + (−2.86 + 1.65i)23-s + (−2.49 − 4.32i)25-s + (−0.972 − 1.68i)29-s + 7.53i·31-s + (0.0474 + 0.0274i)35-s − 1.60i·37-s + (−5.51 + 9.55i)41-s + (−6.35 + 11.0i)43-s + (5.51 − 3.18i)47-s + ⋯
L(s)  = 1  + (0.0108 + 0.00626i)5-s + 0.739·7-s − 0.683i·11-s + (−0.290 + 0.167i)13-s + (−1.37 − 0.795i)17-s + (−0.896 + 0.442i)19-s + (−0.597 + 0.344i)23-s + (−0.499 − 0.865i)25-s + (−0.180 − 0.312i)29-s + 1.35i·31-s + (0.00802 + 0.00463i)35-s − 0.263i·37-s + (−0.861 + 1.49i)41-s + (−0.969 + 1.67i)43-s + (0.804 − 0.464i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.998 - 0.0586i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.998 - 0.0586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05889298651\)
\(L(\frac12)\) \(\approx\) \(0.05889298651\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.90 - 1.93i)T \)
good5 \( 1 + (-0.0242 - 0.0140i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 + 2.26iT - 11T^{2} \)
13 \( 1 + (1.04 - 0.605i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.68 + 3.27i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.86 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.972 + 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.53iT - 31T^{2} \)
37 \( 1 + 1.60iT - 37T^{2} \)
41 \( 1 + (5.51 - 9.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.35 - 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.51 + 3.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.92 + 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.67 + 4.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.233 + 0.403i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.18 + 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.11 - 1.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.85 + 3.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.79 - 2.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.64iT - 83T^{2} \)
89 \( 1 + (1.78 + 3.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.80 - 5.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.237015569396733679349692968503, −8.001262335617580833457826724879, −6.71316378021506287616213745600, −6.34487600640361301816153949280, −5.15253774587990515749486310956, −4.60384390939288537686912247608, −3.64817664778103932704454549548, −2.51663956136452438146410340963, −1.61277889763830021158106795532, −0.01725392564743959940587374473, 1.78349024314315077649565995822, 2.34911200090482105156284555829, 3.84360693341921132615620479300, 4.44320182549100164193886069181, 5.27730265222712734518782754587, 6.17848600621935943353716643779, 7.00036621668839026213789352518, 7.68215180793265946578158181609, 8.532427120532050922353774423046, 9.040006452012101970555820940881

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