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Properties

Label 2-2736-76.75-c1-0-7
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

\(L(1)\) \(\approx\) \(0.8160423952\)
\(L(\frac12)\) \(\approx\) \(0.8160423952\)
\(L(\frac{3}{2})\) 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

−8.871514872890549963940749732656, −8.072300599623585326760040915027, −7.937766777015723840453356052819, −6.70017549660830932168741558557, −5.95608291928898259537972904243, −5.07201820067357646903424277542, −4.27244947173809923125739471483, −3.10033741844014412811112208208, −2.88286478453466400835803529984, −0.907675755888818132126327637972, 0.36715046489676212865619595323, 1.57781907972986124599014416197, 3.35887396043242496314440452845, 3.84953615282049572959409174705, 4.42388596999363770227320574898, 5.29290994191145443650980358238, 6.79477078078023348921198889784, 7.33407916452515660736996730537, 7.66477428574328447304792934713, 8.316461667977841898924924110689

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