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Properties

Label 2-2736-19.11-c1-0-31
Degree $2$
Conductor $2736$
Sign $0.936 + 0.350i$
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  + (0.235 − 0.407i)5-s + 3.30·7-s + 1.47·11-s + (1.41 + 2.45i)13-s + (3.35 − 5.81i)17-s + (0.206 − 4.35i)19-s + (−0.235 − 0.407i)23-s + (2.38 + 4.13i)25-s + (2.48 + 4.30i)29-s − 6.74·31-s + (0.778 − 1.34i)35-s + 4.74·37-s + (−0.250 + 0.434i)41-s + (1.94 − 3.37i)43-s + (−5.95 − 10.3i)47-s + ⋯
L(s)  = 1  + (0.105 − 0.182i)5-s + 1.25·7-s + 0.443·11-s + (0.393 + 0.681i)13-s + (0.814 − 1.41i)17-s + (0.0472 − 0.998i)19-s + (−0.0490 − 0.0849i)23-s + (0.477 + 0.827i)25-s + (0.461 + 0.799i)29-s − 1.21·31-s + (0.131 − 0.227i)35-s + 0.780·37-s + (−0.0391 + 0.0678i)41-s + (0.297 − 0.514i)43-s + (−0.868 − 1.50i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.393967446\)
\(L(\frac12)\) \(\approx\) \(2.393967446\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.206 + 4.35i)T \)
good5 \( 1 + (-0.235 + 0.407i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (-1.41 - 2.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.35 + 5.81i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.235 + 0.407i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.48 - 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + (0.250 - 0.434i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.94 + 3.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.95 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.35 + 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.62 - 6.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.26 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.316 + 0.548i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.88 + 3.27i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.97 + 3.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.41 - 7.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.14T + 83T^{2} \)
89 \( 1 + (-1.47 - 2.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.0293 - 0.0507i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.992131319717660794745202181122, −8.018440861088636561000440285466, −7.24353158149061586874838767666, −6.68346965903973620019977204408, −5.38429527293785200695156287165, −5.03200070295719093606193779533, −4.11126731528939955740894762806, −3.06192224738585658876255105834, −1.90041403357328413741456561714, −0.962741870965517403849428894246, 1.15666693086010835671276383774, 1.98027876713236566087506748135, 3.28036609212485406099793606371, 4.10212037664701258248807392467, 4.95733414980140039471722970044, 5.91917594562541902274465725890, 6.34249653066965849962603675027, 7.76148221740403339371461658689, 7.939189256570541867817819758905, 8.677206298960364646236811558659

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