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Properties

Label 2-2736-76.75-c1-0-8
Degree $2$
Conductor $2736$
Sign $-0.802 - 0.596i$
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  + 3.46i·7-s + 6.92i·13-s + (4 − 1.73i)19-s − 5·25-s − 4·31-s + 6.92i·37-s − 10.3i·43-s − 4.99·49-s − 14·61-s − 16·67-s + 10·73-s + 4·79-s − 23.9·91-s + 13.8i·97-s − 20·103-s + ⋯
L(s)  = 1  + 1.30i·7-s + 1.92i·13-s + (0.917 − 0.397i)19-s − 25-s − 0.718·31-s + 1.13i·37-s − 1.58i·43-s − 0.714·49-s − 1.79·61-s − 1.95·67-s + 1.17·73-s + 0.450·79-s − 2.51·91-s + 1.40i·97-s − 1.97·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.802 - 0.596i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.163259605\)
\(L(\frac12)\) \(\approx\) \(1.163259605\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.258325169173506058864029629449, −8.535455777858778037774105948695, −7.59247474500704955408604414765, −6.80094788714854097119835323625, −6.05477287361117004818924226404, −5.29135597604977596263929489311, −4.47042951013442703450900285351, −3.48251934844891486970631108746, −2.40050699229020987835106063028, −1.62946041223643571654406370624, 0.37413348455242955654809458644, 1.45485945549984179192229020817, 2.95207365975957457267832019900, 3.63030653343448141276192639751, 4.52195966249771805138893023668, 5.50950867430770717917912748299, 6.09806107116109456996088568771, 7.35457138972783893423692290072, 7.59788129209117794402763414394, 8.315596968082283377717599883279

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