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Properties

Label 2-2736-19.7-c1-0-4
Degree $2$
Conductor $2736$
Sign $0.0977 - 0.995i$
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  + (−1 − 1.73i)5-s − 3·7-s − 2·11-s + (0.5 − 0.866i)13-s + (−3 − 5.19i)17-s + (4 + 1.73i)19-s + (−2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (1 − 1.73i)29-s − 7·31-s + (3 + 5.19i)35-s + 37-s + (4 + 6.92i)41-s + (3.5 + 6.06i)43-s + (4 − 6.92i)47-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s − 1.13·7-s − 0.603·11-s + (0.138 − 0.240i)13-s + (−0.727 − 1.26i)17-s + (0.917 + 0.397i)19-s + (−0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (0.185 − 0.321i)29-s − 1.25·31-s + (0.507 + 0.878i)35-s + 0.164·37-s + (0.624 + 1.08i)41-s + (0.533 + 0.924i)43-s + (0.583 − 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5054067470\)
\(L(\frac12)\) \(\approx\) \(0.5054067470\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.254588465403944963934804538998, −8.153649840448480367314574553022, −7.58383624909787730661035929684, −6.77330127275235764807102755320, −5.85158369645341203304735718996, −5.14007424707999011979484774012, −4.26378358551390660447476156966, −3.34086295204750639435431510327, −2.51769483551444735275502002796, −0.945272041674421326246316822670, 0.19730853559321172433349778748, 2.01060275037618941154206401212, 3.06837802004600262315453712001, 3.63887174160624148002397516597, 4.60653433172373397212413852645, 5.78598780566470540757099855452, 6.36477781932498180868876626302, 7.17833761686351194720418511256, 7.68297041940295429103608388902, 8.784259842871944441195996534683

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