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Properties

Label 2-1932-1932.1175-c0-0-0
Degree $2$
Conductor $1932$
Sign $-0.451 - 0.892i$
Analytic cond. $0.964193$
Root an. cond. $0.981933$
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  + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.797 − 0.234i)5-s + (0.415 − 0.909i)6-s + (−0.654 + 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.544 − 0.627i)10-s + (−1.10 + 0.708i)11-s + (0.841 − 0.540i)12-s + (−0.959 + 0.281i)14-s + (−0.118 + 0.822i)15-s + (−0.654 + 0.755i)16-s + (−0.118 + 0.258i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.797 − 0.234i)5-s + (0.415 − 0.909i)6-s + (−0.654 + 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.544 − 0.627i)10-s + (−1.10 + 0.708i)11-s + (0.841 − 0.540i)12-s + (−0.959 + 0.281i)14-s + (−0.118 + 0.822i)15-s + (−0.654 + 0.755i)16-s + (−0.118 + 0.258i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ -0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9507283352\)
\(L(\frac12)\) \(\approx\) \(0.9507283352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 - 0.540i)T \)
3 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good5 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.959 - 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (-0.415 - 0.909i)T^{2} \)
71 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 - 0.540i)T^{2} \)
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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

−9.452012577377949917920162088856, −8.408935594246263702460269377634, −7.78961287275188189193572388676, −7.40788867148193227902000280372, −6.37721829195012445234841361041, −5.69631419009249096842818706346, −5.05247725170090325176667470321, −3.83195841164728548962727629406, −2.95288323315537857025503609122, −1.97486724964418132071811740186, 0.50313827624972600833682024562, 2.98420464403162118895276189908, 3.03804968345166723445124470505, 4.24205639243147142971336811053, 4.78370234236473483867878011045, 5.67936901511054206939366116449, 6.61780456140153089034438130196, 7.39881154050191612715142945359, 8.466902237523432966837646625580, 9.446465144721304786015238099347

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