Properties

Label 2-2736-19.18-c2-0-10
Degree $2$
Conductor $2736$
Sign $-0.846 - 0.533i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.84·5-s + 3.94·7-s − 15.1·11-s − 1.55i·13-s − 2.94·17-s + (−16.0 − 10.1i)19-s − 20.2·23-s + 21.9·25-s − 4.45i·29-s + 3.63i·31-s + 27.0·35-s + 17.6i·37-s + 76.0i·41-s − 26.7·43-s − 48.9·47-s + ⋯
L(s)  = 1  + 1.36·5-s + 0.564·7-s − 1.37·11-s − 0.119i·13-s − 0.173·17-s + (−0.846 − 0.533i)19-s − 0.879·23-s + 0.876·25-s − 0.153i·29-s + 0.117i·31-s + 0.772·35-s + 0.477i·37-s + 1.85i·41-s − 0.621·43-s − 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.846 - 0.533i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.846 - 0.533i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6790539842\)
\(L(\frac12)\) \(\approx\) \(0.6790539842\)
\(L(2)\) 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 \( 1 + (16.0 + 10.1i)T \)
good5 \( 1 - 6.84T + 25T^{2} \)
7 \( 1 - 3.94T + 49T^{2} \)
11 \( 1 + 15.1T + 121T^{2} \)
13 \( 1 + 1.55iT - 169T^{2} \)
17 \( 1 + 2.94T + 289T^{2} \)
23 \( 1 + 20.2T + 529T^{2} \)
29 \( 1 + 4.45iT - 841T^{2} \)
31 \( 1 - 3.63iT - 961T^{2} \)
37 \( 1 - 17.6iT - 1.36e3T^{2} \)
41 \( 1 - 76.0iT - 1.68e3T^{2} \)
43 \( 1 + 26.7T + 1.84e3T^{2} \)
47 \( 1 + 48.9T + 2.20e3T^{2} \)
53 \( 1 - 49.0iT - 2.80e3T^{2} \)
59 \( 1 - 97.9iT - 3.48e3T^{2} \)
61 \( 1 + 105.T + 3.72e3T^{2} \)
67 \( 1 + 129. iT - 4.48e3T^{2} \)
71 \( 1 - 130. iT - 5.04e3T^{2} \)
73 \( 1 + 15.6T + 5.32e3T^{2} \)
79 \( 1 - 113. iT - 6.24e3T^{2} \)
83 \( 1 - 62.5T + 6.88e3T^{2} \)
89 \( 1 + 134. iT - 7.92e3T^{2} \)
97 \( 1 + 116. iT - 9.40e3T^{2} \)
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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

−8.947994191487107361280037790398, −8.198495367222973539105252230751, −7.57912335006764524933339972295, −6.43418167383099446238711013916, −5.95976293889117075217121554016, −5.04643637774681456451789234561, −4.53625407783715466935285814138, −3.03179668699804211614165814330, −2.28359577113474650741824420035, −1.45635668620838897693280777254, 0.13185267270563884268271335206, 1.83964209562744990597472380666, 2.15710360823120919971395382399, 3.38380599347498942512853825756, 4.60149092094940845080247779363, 5.30551281031366038741720126786, 5.94108203723576489398689483217, 6.67254740141008191130260119559, 7.73326292051563864091114867998, 8.289261721196410761688013291910

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