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Properties

Label 2-2736-76.75-c1-0-48
Degree $2$
Conductor $2736$
Sign $i$
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  + 4.35·5-s − 4.35i·7-s − 5i·11-s − 4.35·17-s + 4.35i·19-s − 4i·23-s + 14.0·25-s − 19.0i·35-s − 13.0i·43-s + 13i·47-s − 12.0·49-s − 21.7i·55-s − 15·61-s + 11·73-s − 21.7·77-s + ⋯
L(s)  = 1  + 1.94·5-s − 1.64i·7-s − 1.50i·11-s − 1.05·17-s + 0.999i·19-s − 0.834i·23-s + 2.80·25-s − 3.21i·35-s − 1.99i·43-s + 1.89i·47-s − 1.71·49-s − 2.93i·55-s − 1.92·61-s + 1.28·73-s − 2.48·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.468439797\)
\(L(\frac12)\) \(\approx\) \(2.468439797\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35iT \)
good5 \( 1 - 4.35T + 5T^{2} \)
7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 13.0iT - 43T^{2} \)
47 \( 1 - 13iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.775525495138115540240610581691, −7.923372341857755962970178094588, −6.87387544496115676902571553343, −6.31847859708061915387609853199, −5.72329345799784883718101341847, −4.76786328692634270191123704793, −3.82150356543324644367158578612, −2.81208409348491348703958732225, −1.72862484720391282786684488239, −0.75518234158826439867887008217, 1.73576558408200462915813677538, 2.19599399456952990195144291216, 2.93461895604809677715017313772, 4.73511794364312292464378267073, 5.10067403939386633934636819542, 6.01371754861913096092899387696, 6.48843791692964270617488348650, 7.33800818788570205378967409413, 8.616662461468960036603506179786, 9.186152058103650770582281950237

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