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Properties

Label 1-1932-1932.1811-r0-0-0
Degree $1$
Conductor $1932$
Sign $0.721 + 0.692i$
Analytic cond. $8.97217$
Root an. cond. $8.97217$
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  + (0.995 − 0.0950i)5-s + (0.327 + 0.945i)11-s + (0.959 − 0.281i)13-s + (−0.928 − 0.371i)17-s + (−0.928 + 0.371i)19-s + (0.981 − 0.189i)25-s + (0.142 + 0.989i)29-s + (0.0475 + 0.998i)31-s + (−0.580 + 0.814i)37-s + (0.415 − 0.909i)41-s + (0.841 + 0.540i)43-s + (0.5 + 0.866i)47-s + (0.723 − 0.690i)53-s + (0.415 + 0.909i)55-s + (−0.235 + 0.971i)59-s + ⋯
L(s)  = 1  + (0.995 − 0.0950i)5-s + (0.327 + 0.945i)11-s + (0.959 − 0.281i)13-s + (−0.928 − 0.371i)17-s + (−0.928 + 0.371i)19-s + (0.981 − 0.189i)25-s + (0.142 + 0.989i)29-s + (0.0475 + 0.998i)31-s + (−0.580 + 0.814i)37-s + (0.415 − 0.909i)41-s + (0.841 + 0.540i)43-s + (0.5 + 0.866i)47-s + (0.723 − 0.690i)53-s + (0.415 + 0.909i)55-s + (−0.235 + 0.971i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(8.97217\)
Root analytic conductor: \(8.97217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1932,\ (0:\ ),\ 0.721 + 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.840160020 + 0.7396437490i\)
\(L(\frac12)\) \(\approx\) \(1.840160020 + 0.7396437490i\)
\(L(1)\) \(\approx\) \(1.295032979 + 0.1544417507i\)
\(L(1)\) \(\approx\) \(1.295032979 + 0.1544417507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.995 - 0.0950i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.928 - 0.371i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (-0.580 + 0.814i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.786 + 0.618i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (-0.0475 + 0.998i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.95485598746847017989700641484, −19.0600414894174625538610062873, −18.56862585229112640339097234217, −17.62673306944493119658323696874, −17.14442631107192008069285612997, −16.38157130188635695756604751926, −15.54434640517749034260228808874, −14.77291526889597170153730801859, −13.83165088927126049753595516330, −13.46786999670371005360314289694, −12.76832140264797493038793032724, −11.63255617025841610536333270777, −10.91597488358037150639383106687, −10.40147504023751815058654260977, −9.22266435858083607361188723582, −8.883314499056426243210867496571, −7.999305692810783953483082747217, −6.76662587047996175625206882131, −6.1808303044881996942376747079, −5.657962340754832979806485789337, −4.43896209400894408726285172710, −3.72149963474996501413214355645, −2.554173204781458458460251715110, −1.87994825816638165170014381061, −0.72760415879001635217582328369, 1.17752808106882086922311175339, 1.94330978555822548572668108987, 2.81433482428193994500508930005, 3.95520286449601116765407168497, 4.78373510320993423606960641818, 5.61016241943615296518262848398, 6.51579805220900265488655983471, 6.97188342624429595142875943990, 8.22734679909617439211544503866, 8.9587577429937141197140883082, 9.54928594670903736866693957428, 10.5804091714313219256913490525, 10.88686499285872948495769535156, 12.22310374066724337833603565873, 12.696342168574516668710255792530, 13.55783853374179021772367524240, 14.13131042683909710147409217160, 14.97017118318207242671798478902, 15.72189463549264831291459579512, 16.53188260903507661343121637408, 17.37848384589104271496706516194, 17.86036642643618399465661402223, 18.45553938426310746687039114081, 19.46867758148206677129121067860, 20.22973475798127567426299258583

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