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Properties

Label 1-1832-1832.987-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.157 - 0.987i$
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  + (0.716 − 0.697i)3-s + (0.298 + 0.954i)5-s + (0.350 − 0.936i)7-s + (0.0275 − 0.999i)9-s + (0.789 − 0.614i)11-s + (−0.677 + 0.735i)13-s + (0.879 + 0.475i)15-s + (−0.677 + 0.735i)17-s + (0.975 − 0.218i)19-s + (−0.401 − 0.915i)21-s + (−0.191 − 0.981i)23-s + (−0.821 + 0.569i)25-s + (−0.677 − 0.735i)27-s + (0.635 + 0.771i)29-s + (0.137 + 0.990i)31-s + ⋯
L(s)  = 1  + (0.716 − 0.697i)3-s + (0.298 + 0.954i)5-s + (0.350 − 0.936i)7-s + (0.0275 − 0.999i)9-s + (0.789 − 0.614i)11-s + (−0.677 + 0.735i)13-s + (0.879 + 0.475i)15-s + (−0.677 + 0.735i)17-s + (0.975 − 0.218i)19-s + (−0.401 − 0.915i)21-s + (−0.191 − 0.981i)23-s + (−0.821 + 0.569i)25-s + (−0.677 − 0.735i)27-s + (0.635 + 0.771i)29-s + (0.137 + 0.990i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.987207045 - 2.328683373i\)
\(L(\frac12)\) \(\approx\) \(1.987207045 - 2.328683373i\)
\(L(1)\) \(\approx\) \(1.430224343 - 0.4396978978i\)
\(L(1)\) \(\approx\) \(1.430224343 - 0.4396978978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.716 - 0.697i)T \)
5 \( 1 + (0.298 + 0.954i)T \)
7 \( 1 + (0.350 - 0.936i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.677 + 0.735i)T \)
17 \( 1 + (-0.677 + 0.735i)T \)
19 \( 1 + (0.975 - 0.218i)T \)
23 \( 1 + (-0.191 - 0.981i)T \)
29 \( 1 + (0.635 + 0.771i)T \)
31 \( 1 + (0.137 + 0.990i)T \)
37 \( 1 + (0.592 - 0.805i)T \)
41 \( 1 + (-0.851 + 0.523i)T \)
43 \( 1 + (-0.401 - 0.915i)T \)
47 \( 1 + (0.0275 - 0.999i)T \)
53 \( 1 + (-0.245 - 0.969i)T \)
59 \( 1 + (0.592 + 0.805i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (-0.851 - 0.523i)T \)
71 \( 1 + (0.926 - 0.376i)T \)
73 \( 1 + (0.821 - 0.569i)T \)
79 \( 1 + (0.635 - 0.771i)T \)
83 \( 1 + (0.993 - 0.110i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.904 - 0.426i)T \)
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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

−20.25885604597629689464365645730, −19.727291122142718490538148445127, −18.82832742711134688495983032982, −17.77308836855718659607882353660, −17.30793511482808888639998204280, −16.373543079482453127831633871317, −15.569234083088758908829650665834, −15.20808972258578555683623054044, −14.28365784485259538233678716655, −13.59744858591154719541476399548, −12.80884499114872874879981032918, −11.88349760791056730304609439181, −11.42210541672589964059501009438, −9.92966985531122077695512940876, −9.62290237671186643770168619982, −9.00610015122891705631159086935, −8.12604750387315942525018203148, −7.57143164078544576861189018793, −6.20537648327689319934951461174, −5.207435656881440529392566706178, −4.8247190298278046014786187668, −3.913689495391795239642904850217, −2.76869741727017954708794038265, −2.099534714134462641850747697146, −1.06594569003455428266915596967, 0.489820656012742825974196685680, 1.5535761203682955511421672568, 2.27627435108973292742664802521, 3.33296955348755279572969143218, 3.88882756069355124790429390608, 5.03128022367943188626959764776, 6.511031367033764874268494602429, 6.66819131899463785948858132725, 7.44997099656093531718301836187, 8.35987912729231225984387625804, 9.10142080435664332349224519433, 10.00842011631568817899779035611, 10.74962718365356990880931661071, 11.59313844470957573946912090067, 12.2910114101037044765055871154, 13.46368945387247300323418783464, 13.83363428546970910388135396414, 14.54398071226673461630509784495, 14.88135876187098550494697472621, 16.162715191190182641132013791309, 16.97562351487356202956167423067, 17.78090880080950058097452003696, 18.24458325115791108010575089421, 19.18320962593582670540011770293, 19.7003953395789089049726855343

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