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Properties

Label 2-2736-57.50-c1-0-26
Degree $2$
Conductor $2736$
Sign $-0.0622 + 0.998i$
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  + (−0.557 + 0.321i)5-s − 2.15·7-s − 2.27i·11-s + (−0.313 − 0.181i)13-s + (−1.40 + 0.813i)17-s + (2.64 + 3.46i)19-s + (5.64 + 3.25i)23-s + (−2.29 + 3.97i)25-s + (0.968 − 1.67i)29-s − 6.67i·31-s + (1.19 − 0.692i)35-s − 3.30i·37-s + (−3.49 − 6.05i)41-s + (−2.71 − 4.70i)43-s + (7.69 + 4.44i)47-s + ⋯
L(s)  = 1  + (−0.249 + 0.143i)5-s − 0.813·7-s − 0.684i·11-s + (−0.0869 − 0.0502i)13-s + (−0.341 + 0.197i)17-s + (0.606 + 0.795i)19-s + (1.17 + 0.679i)23-s + (−0.458 + 0.794i)25-s + (0.179 − 0.311i)29-s − 1.19i·31-s + (0.202 − 0.117i)35-s − 0.544i·37-s + (−0.545 − 0.945i)41-s + (−0.414 − 0.717i)43-s + (1.12 + 0.648i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0622 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0622 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0622 + 0.998i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.0622 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9499210396\)
\(L(\frac12)\) \(\approx\) \(0.9499210396\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 3.46i)T \)
good5 \( 1 + (0.557 - 0.321i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.15T + 7T^{2} \)
11 \( 1 + 2.27iT - 11T^{2} \)
13 \( 1 + (0.313 + 0.181i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.40 - 0.813i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.64 - 3.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.968 + 1.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.67iT - 31T^{2} \)
37 \( 1 + 3.30iT - 37T^{2} \)
41 \( 1 + (3.49 + 6.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 + 4.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.69 - 4.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.96 + 3.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.93 + 5.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.91 - 6.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 + 6.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.60 + 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.63 + 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.77 - 1.60i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 + (-8.16 + 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.37 - 4.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.831945306694003138369664271071, −7.62665344020950955763681527356, −7.33360678205424658876233000000, −6.15642883757777633749393745627, −5.76433852595363110700934984180, −4.67409595808202992769283626258, −3.56250597106241638539310363284, −3.15584881133971987501090113909, −1.80755041444206623551251189586, −0.34304620512433182976511693944, 1.10984711306556239010798920903, 2.57307869871087670363892383624, 3.24248028480935169129801803279, 4.44234135737325194617124743082, 4.94787302647308945952909261517, 6.05838981098087394204404606274, 6.91094687818951303846916273181, 7.25718083390552252194377639953, 8.435454500323395085386392584354, 8.952416665443771744244056595225

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