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Properties

Label 2-1872-117.70-c0-0-1
Degree $2$
Conductor $1872$
Sign $0.800 - 0.599i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1.36 − 0.366i)7-s + (−0.499 + 0.866i)9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)13-s − i·17-s + (1 − i)19-s + (1 + 0.999i)21-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s − 0.999·27-s + (1 + 0.999i)33-s + (1 + i)37-s + (−0.866 − 0.499i)39-s + (−1.36 − 0.366i)41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (1.36 − 0.366i)7-s + (−0.499 + 0.866i)9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)13-s − i·17-s + (1 − i)19-s + (1 + 0.999i)21-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s − 0.999·27-s + (1 + 0.999i)33-s + (1 + i)37-s + (−0.866 − 0.499i)39-s + (−1.36 − 0.366i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1825, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581314950\)
\(L(\frac12)\) \(\approx\) \(1.581314950\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−9.537180488329848088998119403018, −8.845724661098549185477509225712, −7.938212589205832580112158427807, −7.41184528901657582291063166299, −6.31281513578766730257923791513, −5.07938026219760773772115606422, −4.63634961670779148986093213389, −3.80963058761701487157235716757, −2.70618525650891556390506239541, −1.52421086606574597006503406665, 1.50667707705467885057641092283, 2.02776429859673837017409988876, 3.42482540758769838936075216008, 4.31482436325630064132513742793, 5.45644137040267498218810802556, 6.18076058543716551068180160290, 7.14061249417357075701763428943, 8.026477024728232844472920215938, 8.169956766363443626170422291972, 9.308089923509075376140031930325

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