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Properties

Label 2-1872-13.3-c1-0-10
Degree $2$
Conductor $1872$
Sign $0.0128 - 0.999i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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  + 2.64·5-s + (−1 + 1.73i)7-s + (2.64 + 4.58i)11-s + (3.5 − 0.866i)13-s + (−3.96 + 6.87i)17-s + (−3 + 5.19i)19-s + (−2.64 − 4.58i)23-s + 2.00·25-s + (−1.32 − 2.29i)29-s − 4·31-s + (−2.64 + 4.58i)35-s + (1.5 + 2.59i)37-s + (−3.96 − 6.87i)41-s + (−1 + 1.73i)43-s − 5.29·47-s + ⋯
L(s)  = 1  + 1.18·5-s + (−0.377 + 0.654i)7-s + (0.797 + 1.38i)11-s + (0.970 − 0.240i)13-s + (−0.962 + 1.66i)17-s + (−0.688 + 1.19i)19-s + (−0.551 − 0.955i)23-s + 0.400·25-s + (−0.245 − 0.425i)29-s − 0.718·31-s + (−0.447 + 0.774i)35-s + (0.246 + 0.427i)37-s + (−0.619 − 1.07i)41-s + (−0.152 + 0.264i)43-s − 0.771·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892280269\)
\(L(\frac12)\) \(\approx\) \(1.892280269\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.64 - 4.58i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.96 - 6.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.32 + 2.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.96 + 6.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.29T + 47T^{2} \)
53 \( 1 - 7.93T + 53T^{2} \)
59 \( 1 + (-5.29 + 9.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.64 + 4.58i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.366681713218821500267760760835, −8.783289748835940229504284195862, −8.043692128927019105012783825477, −6.71805580433408657966446031537, −6.19916373103312667890364821080, −5.71633226273218670214068623076, −4.41615029093737791377641634983, −3.68587304071269763456484824417, −2.08537682249800111210587823652, −1.78853177239135530188562841383, 0.66498038271640138476457699571, 1.91245677924523517405062517560, 3.09154547222709204131378356784, 3.95855055913473349736238255091, 5.07473436197625952212869228780, 5.97836891105535028977190037435, 6.56938740075092842921131018307, 7.22438660333770069759719162956, 8.544039743952668585153081789831, 9.124557144184054774041998062267

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