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Properties

Label 1-1932-1932.1739-r0-0-0
Degree $1$
Conductor $1932$
Sign $-0.940 - 0.340i$
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  + (−0.235 + 0.971i)5-s + (−0.0475 + 0.998i)11-s + (0.654 − 0.755i)13-s + (−0.580 + 0.814i)17-s + (−0.580 − 0.814i)19-s + (−0.888 − 0.458i)25-s + (−0.415 − 0.909i)29-s + (−0.786 + 0.618i)31-s + (−0.723 + 0.690i)37-s + (−0.959 + 0.281i)41-s + (−0.142 + 0.989i)43-s + (0.5 − 0.866i)47-s + (−0.327 + 0.945i)53-s + (−0.959 − 0.281i)55-s + (−0.981 − 0.189i)59-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)5-s + (−0.0475 + 0.998i)11-s + (0.654 − 0.755i)13-s + (−0.580 + 0.814i)17-s + (−0.580 − 0.814i)19-s + (−0.888 − 0.458i)25-s + (−0.415 − 0.909i)29-s + (−0.786 + 0.618i)31-s + (−0.723 + 0.690i)37-s + (−0.959 + 0.281i)41-s + (−0.142 + 0.989i)43-s + (0.5 − 0.866i)47-s + (−0.327 + 0.945i)53-s + (−0.959 − 0.281i)55-s + (−0.981 − 0.189i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.940 - 0.340i$
Analytic conductor: \(8.97217\)
Root analytic conductor: \(8.97217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1932,\ (0:\ ),\ -0.940 - 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04614806616 + 0.2630486238i\)
\(L(\frac12)\) \(\approx\) \(-0.04614806616 + 0.2630486238i\)
\(L(1)\) \(\approx\) \(0.7819054844 + 0.2015288532i\)
\(L(1)\) \(\approx\) \(0.7819054844 + 0.2015288532i\)

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 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.580 + 0.814i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.786 + 0.618i)T \)
37 \( 1 + (-0.723 + 0.690i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.327 + 0.945i)T \)
59 \( 1 + (-0.981 - 0.189i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.995 - 0.0950i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−19.64948906718890657866984526488, −18.840238682793369791855961232875, −18.365061200189013190059839927860, −17.2357893225503299328207720439, −16.61756129139457066305155256078, −16.08682817818173116722146869198, −15.449425019473015955591030737161, −14.29781455240131140450511885240, −13.74032111874561153704053758182, −12.96358065668890142711384285541, −12.26942154143619231437765167792, −11.38945073688044544287198226779, −10.90348728909124362524022749146, −9.76129787812724679622368009677, −8.82756744997087746291618768894, −8.6244515839950324152292436045, −7.57944382917536933677227204073, −6.67147598223296826017855186683, −5.73959275586370711424860075811, −5.089769701641079983556213244736, −4.04542199882121748325958199268, −3.51664353238059739392805371910, −2.155135189653255564408269189732, −1.297543808049630016206702601430, −0.09032877159832385129509301941, 1.59551769632789295435628692907, 2.448854422962146589598920540096, 3.36565456807887966954586945470, 4.1471081879535970588265819075, 5.069776031274549953558160389010, 6.177889769388811503138304258867, 6.727040499019683881871730501216, 7.56587548268884099572240324378, 8.30602866876239370858375513727, 9.21762062057067382084384707898, 10.22221849684520326591393356719, 10.68709014382951398660904779427, 11.41636084482548661876814693276, 12.31641408789807714516668126240, 13.11675120941114680477636888130, 13.75321961342447513166331588202, 14.87039779074040488602213455670, 15.19770550443677915658572762438, 15.76079444567533485963902611339, 16.97638305240209798475234527956, 17.62067288536374671269583616691, 18.21431385530147596931517998330, 18.92627301998059717335513852100, 19.78403911370707558557981169973, 20.24294553134294300299158187207

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