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Properties

Label 2-1932-1932.167-c0-0-1
Degree $2$
Conductor $1932$
Sign $-0.117 + 0.993i$
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  + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.10 + 0.708i)5-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.186 + 1.29i)10-s + (−0.118 − 0.258i)11-s + (0.415 + 0.909i)12-s + (0.841 + 0.540i)14-s + (1.25 − 0.368i)15-s + (−0.142 + 0.989i)16-s + (1.25 − 1.45i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.10 + 0.708i)5-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.186 + 1.29i)10-s + (−0.118 − 0.258i)11-s + (0.415 + 0.909i)12-s + (0.841 + 0.540i)14-s + (1.25 − 0.368i)15-s + (−0.142 + 0.989i)16-s + (1.25 − 1.45i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6862118413\)
\(L(\frac12)\) \(\approx\) \(0.6862118413\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good5 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−9.504139970328746988082150182433, −8.340278420377678214490543756131, −7.62712351260864216400298559148, −6.53860018476954574404203438131, −5.94638103815493557144031101080, −4.97621569137776519581086840815, −4.31084545173007624187658811297, −3.10432533447653813925428557613, −2.43239623179537229907956906945, −0.67137160689429899237721530686, 1.00464703120048487672512564684, 3.52786470512635366650525230752, 4.14712961666997962439036124636, 4.58794490605182489147373027588, 5.70029274001767148487134707057, 6.27299170741359076171514256428, 7.28441076106415956715473301366, 7.904072864542173234923136326675, 8.406051451894832194573922243179, 9.727497151359079818993560050280

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