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Properties

Label 1-1832-1832.13-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.936 - 0.351i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (0.0825 + 0.996i)3-s + (0.245 − 0.969i)5-s + (0.837 − 0.546i)7-s + (−0.986 + 0.164i)9-s + (−0.677 − 0.735i)11-s + (−0.969 − 0.245i)13-s + (0.986 + 0.164i)15-s + (0.245 − 0.969i)17-s + (−0.245 − 0.969i)19-s + (0.614 + 0.789i)21-s + (0.915 − 0.401i)23-s + (−0.879 − 0.475i)25-s + (−0.245 − 0.969i)27-s + (0.837 − 0.546i)29-s + (0.735 − 0.677i)31-s + ⋯
L(s)  = 1  + (0.0825 + 0.996i)3-s + (0.245 − 0.969i)5-s + (0.837 − 0.546i)7-s + (−0.986 + 0.164i)9-s + (−0.677 − 0.735i)11-s + (−0.969 − 0.245i)13-s + (0.986 + 0.164i)15-s + (0.245 − 0.969i)17-s + (−0.245 − 0.969i)19-s + (0.614 + 0.789i)21-s + (0.915 − 0.401i)23-s + (−0.879 − 0.475i)25-s + (−0.245 − 0.969i)27-s + (0.837 − 0.546i)29-s + (0.735 − 0.677i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.936 - 0.351i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.936 - 0.351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2150368776 - 1.185267463i\)
\(L(\frac12)\) \(\approx\) \(0.2150368776 - 1.185267463i\)
\(L(1)\) \(\approx\) \(1.016474378 - 0.1787713049i\)
\(L(1)\) \(\approx\) \(1.016474378 - 0.1787713049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.0825 + 0.996i)T \)
5 \( 1 + (0.245 - 0.969i)T \)
7 \( 1 + (0.837 - 0.546i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (-0.969 - 0.245i)T \)
17 \( 1 + (0.245 - 0.969i)T \)
19 \( 1 + (-0.245 - 0.969i)T \)
23 \( 1 + (0.915 - 0.401i)T \)
29 \( 1 + (0.837 - 0.546i)T \)
31 \( 1 + (0.735 - 0.677i)T \)
37 \( 1 + (-0.789 + 0.614i)T \)
41 \( 1 + (-0.164 + 0.986i)T \)
43 \( 1 + (-0.789 + 0.614i)T \)
47 \( 1 + (0.164 + 0.986i)T \)
53 \( 1 + (0.0825 + 0.996i)T \)
59 \( 1 + (0.614 - 0.789i)T \)
61 \( 1 + (0.986 - 0.164i)T \)
67 \( 1 + (-0.164 - 0.986i)T \)
71 \( 1 + (0.677 - 0.735i)T \)
73 \( 1 + (0.475 - 0.879i)T \)
79 \( 1 + (-0.837 - 0.546i)T \)
83 \( 1 + (-0.789 - 0.614i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.879 - 0.475i)T \)
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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

−20.19207872858210889079388815281, −19.22171017262025542907872578697, −18.90353921906906179670718194740, −18.08235821287188217239915114386, −17.52616027576098620967514678813, −17.05211444695932682835942679381, −15.64168256856521501282588720466, −14.81049866923378542494505951744, −14.5176497082037229616677218788, −13.73516957290754807574622727011, −12.72455208426464568069096729843, −12.18526348851291403444452862746, −11.48862559683550041983462969017, −10.511464652581938364728989123538, −9.98058708161715013092632332832, −8.66028957344385529846488978117, −8.149076900686358220018802041, −7.11198443689300553224123799484, −6.89475493994047110796632705641, −5.61782370896062578670942423107, −5.19259077121631670261324968881, −3.79777186368594369555812229, −2.6931109559620439459900110285, −2.12394704418233042153216781970, −1.398615932096837430328668621573, 0.22601620860185812008962321261, 0.9511426860508346378556450383, 2.42333645998303384513738307462, 3.10396750820725442569180461715, 4.464529948688911119801476573435, 4.842468950700993372226316283024, 5.33940442344453394032256366432, 6.52658497406886638996450491358, 7.80750864845930973603692968636, 8.272085411391316908810332084898, 9.14283900017995957038572184750, 9.830442271227441590332279357, 10.57337926393609372273751842684, 11.365947286471996324406154105, 11.99978946197569927052410768780, 13.17922915688090355406958161245, 13.72420868674238870257645352107, 14.48909147614373336673659289850, 15.34927490190709839810017018851, 15.95398604146667230110717374256, 16.83626945776700663298675563480, 17.172969150255551294396582810499, 17.92443964258199657815369567085, 19.08448235550543472750923370348, 19.91509616859091881713841819211

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