Properties

Label 2-2736-12.11-c1-0-0
Degree $2$
Conductor $2736$
Sign $-0.908 + 0.418i$
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  + 2.30i·5-s + 4.59i·7-s − 3.79·11-s − 2.59·13-s − 3.13i·17-s + i·19-s + 3.59·23-s − 0.333·25-s + 7.15i·29-s + 6.83i·31-s − 10.6·35-s − 5.26·37-s − 10.9i·41-s + 5.05i·43-s + 2.74·47-s + ⋯
L(s)  = 1  + 1.03i·5-s + 1.73i·7-s − 1.14·11-s − 0.719·13-s − 0.761i·17-s + 0.229i·19-s + 0.750·23-s − 0.0666·25-s + 1.32i·29-s + 1.22i·31-s − 1.79·35-s − 0.864·37-s − 1.71i·41-s + 0.771i·43-s + 0.400·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.908 + 0.418i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7108077161\)
\(L(\frac12)\) \(\approx\) \(0.7108077161\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30iT - 5T^{2} \)
7 \( 1 - 4.59iT - 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
17 \( 1 + 3.13iT - 17T^{2} \)
23 \( 1 - 3.59T + 23T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 - 6.83iT - 31T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 - 5.05iT - 43T^{2} \)
47 \( 1 - 2.74T + 47T^{2} \)
53 \( 1 + 0.137iT - 53T^{2} \)
59 \( 1 + 3.11T + 59T^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 + 2.53iT - 67T^{2} \)
71 \( 1 + 4.05T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 - 7.81T + 83T^{2} \)
89 \( 1 + 6.79iT - 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−8.992438274447017358540156783792, −8.763526397327386711193355361150, −7.56520196936671314587071291760, −7.08523992589234713161792849716, −6.17486254463738451973550028525, −5.27614983729960616268907114022, −4.93667814477818082912402832490, −3.15931247349540774059351384282, −2.85591510485940399136932214858, −1.94794173051809876866867585871, 0.23424487555399625421027024879, 1.21185952821603120024399430978, 2.52586062510770099970894512682, 3.74237181723847465385294811692, 4.52965206838989196858735749277, 5.03508285351411864512402072714, 6.06073234017182229705744294765, 7.06349674028485914406545872019, 7.72265138612458652161705263467, 8.203965566798715427704197493130

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