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Properties

Label 2-2736-19.7-c1-0-22
Degree $2$
Conductor $2736$
Sign $0.996 - 0.0791i$
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  + (0.524 + 0.908i)5-s + 3.44·7-s − 5.71·11-s + (0.5 − 0.866i)13-s + (−1.04 − 1.81i)17-s + (−1 + 4.24i)19-s + (1.80 − 3.13i)23-s + (1.94 − 3.37i)25-s + (3.61 − 6.26i)29-s + 9.44·31-s + (1.80 + 3.13i)35-s + 3.89·37-s + (4.66 + 8.08i)41-s + (3.17 + 5.49i)43-s + (4.66 − 8.08i)47-s + ⋯
L(s)  = 1  + (0.234 + 0.406i)5-s + 1.30·7-s − 1.72·11-s + (0.138 − 0.240i)13-s + (−0.254 − 0.440i)17-s + (−0.229 + 0.973i)19-s + (0.377 − 0.653i)23-s + (0.389 − 0.675i)25-s + (0.672 − 1.16i)29-s + 1.69·31-s + (0.305 + 0.529i)35-s + 0.640·37-s + (0.729 + 1.26i)41-s + (0.484 + 0.838i)43-s + (0.681 − 1.17i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.996 - 0.0791i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.996 - 0.0791i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.077924441\)
\(L(\frac12)\) \(\approx\) \(2.077924441\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 4.24i)T \)
good5 \( 1 + (-0.524 - 0.908i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.04 + 1.81i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.80 + 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.61 + 6.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.44T + 31T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 + (-4.66 - 8.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.17 - 5.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.66 + 8.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.524 - 0.908i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.90 - 6.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.174 + 0.301i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.61 - 6.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.174 - 0.301i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-2.62 + 4.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.55 + 2.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.506064179042874118874862641825, −8.087984677708042862385027597052, −7.57562994780914254812945045729, −6.45725851811339136164033869333, −5.75251145170744417583762593985, −4.82554612200668746677473015756, −4.35350765494776496698712036133, −2.77259430325079765155271929683, −2.37748991150016003630511826696, −0.899301256752125751652623653605, 0.939313067009083076818473614611, 2.09110111613293957116582816196, 2.93026103531456084452607152311, 4.30410189039580593564791226446, 5.04062629827762382642804352550, 5.40269652385194825946901173135, 6.55913255773564249669009941123, 7.50406872905355018591864133372, 8.053097935666737260189998951325, 8.747582563612049552982467764116

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