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Properties

Label 2-1932-1932.1103-c0-0-1
Degree $2$
Conductor $1932$
Sign $0.997 + 0.0633i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.499 + 0.866i)12-s + 13-s + (0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + (−0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.499 + 0.866i)12-s + 13-s + (0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + (−0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ 0.997 + 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031433220\)
\(L(\frac12)\) \(\approx\) \(1.031433220\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.348469412171662226053862214595, −8.804473762425185969933434891634, −7.934589213913128451516090187343, −6.97638633022312829848577915797, −6.41220672214340490363883322691, −5.75007413247898201894338976888, −3.79084577063819999125368959232, −3.21088136361146072058980669549, −2.38747872496154515846411295967, −1.63515571570651923798839333895, 0.877419906271223670555263370558, 2.39995114571498855237091265288, 3.90762069769409550200996501218, 4.70787543664036344583237099487, 5.32284049238625298217355908503, 6.18173592590882054503407882343, 7.02789286704396963138188510682, 8.142182093655252372791655892698, 8.875644054281386625284411898889, 9.168112115948114031326332424533

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