Properties

Label 2-2736-12.11-c1-0-9
Degree $2$
Conductor $2736$
Sign $0.0917 - 0.995i$
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  + 1.11i·5-s − 1.19i·7-s − 3.72·11-s + 0.113·13-s − 2.89i·17-s + i·19-s − 1.35·23-s + 3.76·25-s − 0.860i·29-s + 9.01i·31-s + 1.32·35-s + 5.64·37-s + 10.5i·41-s + 2.47i·43-s + 5.05·47-s + ⋯
L(s)  = 1  + 0.496i·5-s − 0.451i·7-s − 1.12·11-s + 0.0314·13-s − 0.701i·17-s + 0.229i·19-s − 0.282·23-s + 0.753·25-s − 0.159i·29-s + 1.61i·31-s + 0.224·35-s + 0.927·37-s + 1.64i·41-s + 0.376i·43-s + 0.737·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0917 - 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0917 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243662266\)
\(L(\frac12)\) \(\approx\) \(1.243662266\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 1.11iT - 5T^{2} \)
7 \( 1 + 1.19iT - 7T^{2} \)
11 \( 1 + 3.72T + 11T^{2} \)
13 \( 1 - 0.113T + 13T^{2} \)
17 \( 1 + 2.89iT - 17T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + 0.860iT - 29T^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 - 2.47iT - 43T^{2} \)
47 \( 1 - 5.05T + 47T^{2} \)
53 \( 1 - 9.51iT - 53T^{2} \)
59 \( 1 + 8.73T + 59T^{2} \)
61 \( 1 + 8.52T + 61T^{2} \)
67 \( 1 - 0.857iT - 67T^{2} \)
71 \( 1 - 1.91T + 71T^{2} \)
73 \( 1 - 3.06T + 73T^{2} \)
79 \( 1 - 7.62iT - 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−9.032371729200035984806820709917, −8.082111475263518360793136161061, −7.52446625947051597318464207526, −6.79837043120967775006374194260, −5.98160273039316019918939404279, −5.05681207748264119600299151206, −4.35014787402722000493919275386, −3.14992680362851204589400083783, −2.58949098496920594021628931068, −1.12921814110593874558285248638, 0.43827099016146855280787380230, 1.93870892432343067798098003709, 2.78044211009043130047685665023, 3.92043982149117608625070152447, 4.78487867124417079337192974376, 5.57736476527135846541524050570, 6.16117133181170938509649946619, 7.29826048557592997413361096712, 7.920278741954730877942274456192, 8.685494347319332798080619166878

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