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Properties

Label 2-2736-57.8-c1-0-29
Degree $2$
Conductor $2736$
Sign $-0.525 + 0.850i$
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  + (2.90 + 1.67i)5-s − 3.54·7-s + 0.251i·11-s + (−4.29 + 2.47i)13-s + (−3.11 − 1.80i)17-s + (−4.15 + 1.30i)19-s + (3.89 − 2.24i)23-s + (3.11 + 5.39i)25-s + (−2.27 − 3.94i)29-s − 5.28i·31-s + (−10.2 − 5.94i)35-s − 6.47i·37-s + (4.96 − 8.59i)41-s + (3.38 − 5.86i)43-s + (−8.96 + 5.17i)47-s + ⋯
L(s)  = 1  + (1.29 + 0.749i)5-s − 1.34·7-s + 0.0758i·11-s + (−1.18 + 0.687i)13-s + (−0.756 − 0.436i)17-s + (−0.954 + 0.298i)19-s + (0.811 − 0.468i)23-s + (0.622 + 1.07i)25-s + (−0.423 − 0.732i)29-s − 0.949i·31-s + (−1.73 − 1.00i)35-s − 1.06i·37-s + (0.774 − 1.34i)41-s + (0.516 − 0.894i)43-s + (−1.30 + 0.754i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.525 + 0.850i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4872087856\)
\(L(\frac12)\) \(\approx\) \(0.4872087856\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.15 - 1.30i)T \)
good5 \( 1 + (-2.90 - 1.67i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 - 0.251iT - 11T^{2} \)
13 \( 1 + (4.29 - 2.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.11 + 1.80i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.89 + 2.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.27 + 3.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.28iT - 31T^{2} \)
37 \( 1 + 6.47iT - 37T^{2} \)
41 \( 1 + (-4.96 + 8.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.38 + 5.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.96 - 5.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.217 - 0.377i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.06 + 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.46 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.273 - 0.157i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.10 - 7.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.356 + 0.616i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.57 - 4.37i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.31iT - 83T^{2} \)
89 \( 1 + (-1.20 - 2.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.36 - 5.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.931976671526481522526012882452, −7.56077958192782030322484629776, −6.83461934753328091030618174200, −6.37777466764036584023313309279, −5.70865215019579924605996504148, −4.65398176601540912322967366204, −3.65105560044184303287279447353, −2.42375939154427181056733273816, −2.24282402541026824077935994643, −0.14415779199146435884954365668, 1.35744811628746641401132871095, 2.51404763869483719770650899087, 3.22593975581148959624141851298, 4.60059914154346590362554429887, 5.18694343714887091137953398410, 6.13587460034815461906383437674, 6.55937893564218853911997503082, 7.48233650296590587831198655598, 8.618534100457015501364752477166, 9.135597592838912107813090436215

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