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Properties

Label 2-2736-19.7-c1-0-1
Degree $2$
Conductor $2736$
Sign $-0.890 - 0.455i$
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  + (−0.412 − 0.715i)5-s + 0.703·7-s + 1.17·11-s + (−2.45 + 4.25i)13-s + (−0.296 − 0.514i)17-s + (−3.07 + 3.08i)19-s + (−3.85 + 6.67i)23-s + (2.15 − 3.73i)25-s + (4.73 − 8.20i)29-s − 4.26·31-s + (−0.290 − 0.502i)35-s − 7.53·37-s + (−5.03 − 8.71i)41-s + (−1.83 − 3.18i)43-s + (0.877 − 1.51i)47-s + ⋯
L(s)  = 1  + (−0.184 − 0.319i)5-s + 0.265·7-s + 0.354·11-s + (−0.681 + 1.17i)13-s + (−0.0719 − 0.124i)17-s + (−0.706 + 0.707i)19-s + (−0.803 + 1.39i)23-s + (0.431 − 0.747i)25-s + (0.879 − 1.52i)29-s − 0.766·31-s + (−0.0490 − 0.0849i)35-s − 1.23·37-s + (−0.786 − 1.36i)41-s + (−0.280 − 0.485i)43-s + (0.127 − 0.221i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.890 - 0.455i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4030940677\)
\(L(\frac12)\) \(\approx\) \(0.4030940677\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.07 - 3.08i)T \)
good5 \( 1 + (0.412 + 0.715i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.703T + 7T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (2.45 - 4.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.296 + 0.514i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.85 - 6.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.73 + 8.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.26T + 31T^{2} \)
37 \( 1 + 7.53T + 37T^{2} \)
41 \( 1 + (5.03 + 8.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.83 + 3.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.877 + 1.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.46 - 4.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.15 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.10 - 5.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.26 - 9.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.20 - 7.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.151 + 0.262i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.221 - 0.383i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.41T + 83T^{2} \)
89 \( 1 + (-6.55 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.747 - 1.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.987269036621267559333814015974, −8.517135310516562063219790032706, −7.60451495696809677905025797361, −6.93725273012476548334705319945, −6.09286164136073171243320004004, −5.23925756129129491587448175704, −4.30571193621799318507566000711, −3.79518106681889579938298029086, −2.37394323963968318928498304286, −1.54138388333231051778694811563, 0.12385078964316755803598650438, 1.64635907523968466864827403603, 2.81614988672262703389658356834, 3.52885793977938035653255616763, 4.76087407287881988812654087510, 5.16881338083422019699254591028, 6.46652275896794322249377064144, 6.82057769886333306571492579383, 7.88246656142929254875583905709, 8.384351378802625292241574355850

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