Properties

Label 2-2736-171.94-c0-0-0
Degree $2$
Conductor $2736$
Sign $-0.984 + 0.173i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + i·19-s + (−0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s i·27-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s − 0.999·35-s + ⋯
L(s)  = 1  + i·3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + i·19-s + (−0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s i·27-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s − 0.999·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6380062153\)
\(L(\frac12)\) \(\approx\) \(0.6380062153\)
\(L(1)\) 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 \( 1 - iT \)
19 \( 1 - iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.398661319432373077377460821290, −8.715400387625001276348009738225, −7.86163965186582158886816316506, −7.42587666762991906151973686657, −5.98822208701618526537470619763, −5.60472589354089515552422742235, −4.78800305526582150374618692628, −3.62834184681690635374699166656, −3.14286806131684989039542006484, −2.13322146397854402876433217986, 0.38505450821750000300779799779, 1.69968036495218846029467296794, 2.55109123797607687912030611546, 3.99158741392798706638315752250, 4.72725168396198014012369131786, 5.34305900732435903802360748982, 6.59640206573019765371020053732, 7.28942498113989425927474012080, 7.66657356528976352281762309554, 8.495624938794389860622123673557

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