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Properties

Label 2-2736-76.75-c1-0-31
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 − 3.46i·11-s − 3.46i·13-s + 6·17-s + (4 − 1.73i)19-s − 5·25-s − 6.92i·29-s − 10·31-s − 3.46i·37-s + 6.92i·41-s + 10.3i·43-s − 6.92i·47-s − 4.99·49-s − 13.8i·53-s + 12·59-s + ⋯
L(s)  = 1  + 1.30i·7-s − 1.04i·11-s − 0.960i·13-s + 1.45·17-s + (0.917 − 0.397i)19-s − 25-s − 1.28i·29-s − 1.79·31-s − 0.569i·37-s + 1.08i·41-s + 1.58i·43-s − 1.01i·47-s − 0.714·49-s − 1.90i·53-s + 1.56·59-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

\(L(1)\) \(\approx\) \(1.742111701\)
\(L(\frac12)\) \(\approx\) \(1.742111701\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 3.46iT - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 6.92iT - 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−8.590067112621452960464968356750, −8.112699725932733472101803620504, −7.40872140924841305327437322265, −6.19899197619047223189784051360, −5.54401857037134018691675973596, −5.26239202779875142383189519877, −3.69312256337309369945735400237, −3.09558569138143154285089528654, −2.09727542061268540924599918882, −0.65922392756904491271660784896, 1.09623745212119098968117753023, 2.02687537929811228345164510534, 3.56272381534382604322226150596, 3.93938276919960824219396381649, 5.01675021272084989323685143065, 5.71869382657016986079824858758, 6.99540040482353581622120047140, 7.25925218713316289330286749062, 7.900893763137617359634014820226, 9.073208049576535529011561465060

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