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Properties

Label 2-1872-39.35-c0-0-0
Degree $2$
Conductor $1872$
Sign $-0.162 - 0.986i$
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  + 1.41i·5-s + (0.5 + 0.866i)7-s + (−1.22 − 0.707i)11-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)23-s − 1.00·25-s + (−1.22 − 0.707i)29-s − 31-s + (−1.22 + 0.707i)35-s + (0.5 + 0.866i)43-s + 1.41i·47-s + (1.00 − 1.73i)55-s + (−1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯
L(s)  = 1  + 1.41i·5-s + (0.5 + 0.866i)7-s + (−1.22 − 0.707i)11-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)23-s − 1.00·25-s + (−1.22 − 0.707i)29-s − 31-s + (−1.22 + 0.707i)35-s + (0.5 + 0.866i)43-s + 1.41i·47-s + (1.00 − 1.73i)55-s + (−1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ -0.162 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068156475\)
\(L(\frac12)\) \(\approx\) \(1.068156475\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.507306878998669142621319383468, −8.960427535713105880481524403496, −7.88159349524089807796661252591, −7.41801423156554094797001985242, −6.36665264980924535256753070789, −5.76680808716137053579947968946, −4.88437031412233931285100472338, −3.56155695294615152853462184726, −2.82761206328073714035011010263, −1.91860754279898875712100839849, 0.799267715146077860608838832363, 1.98492045971916247638842383840, 3.41293786859977744830943890349, 4.46398627765299169809353965572, 5.10511010416445929708107984835, 5.66849136978678372536973133414, 7.15699950863964623707275142889, 7.61955685396489103432881329702, 8.502153526844189437723539905051, 9.007745355034732865791125510452

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