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Properties

Label 1-1832-1832.117-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.0954 + 0.995i$
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  + (−0.0275 − 0.999i)3-s + (−0.821 − 0.569i)5-s + (0.656 + 0.754i)7-s + (−0.998 + 0.0550i)9-s + (0.245 + 0.969i)11-s + (0.996 + 0.0825i)13-s + (−0.546 + 0.837i)15-s + (−0.0825 + 0.996i)17-s + (−0.904 − 0.426i)19-s + (0.735 − 0.677i)21-s + (0.376 − 0.926i)23-s + (0.350 + 0.936i)25-s + (0.0825 + 0.996i)27-s + (−0.981 + 0.191i)29-s + (0.272 − 0.962i)31-s + ⋯
L(s)  = 1  + (−0.0275 − 0.999i)3-s + (−0.821 − 0.569i)5-s + (0.656 + 0.754i)7-s + (−0.998 + 0.0550i)9-s + (0.245 + 0.969i)11-s + (0.996 + 0.0825i)13-s + (−0.546 + 0.837i)15-s + (−0.0825 + 0.996i)17-s + (−0.904 − 0.426i)19-s + (0.735 − 0.677i)21-s + (0.376 − 0.926i)23-s + (0.350 + 0.936i)25-s + (0.0825 + 0.996i)27-s + (−0.981 + 0.191i)29-s + (0.272 − 0.962i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.0954 + 0.995i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.0954 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4955064244 + 0.5452981241i\)
\(L(\frac12)\) \(\approx\) \(0.4955064244 + 0.5452981241i\)
\(L(1)\) \(\approx\) \(0.8776628895 - 0.1739348519i\)
\(L(1)\) \(\approx\) \(0.8776628895 - 0.1739348519i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.0275 - 0.999i)T \)
5 \( 1 + (-0.821 - 0.569i)T \)
7 \( 1 + (0.656 + 0.754i)T \)
11 \( 1 + (0.245 + 0.969i)T \)
13 \( 1 + (0.996 + 0.0825i)T \)
17 \( 1 + (-0.0825 + 0.996i)T \)
19 \( 1 + (-0.904 - 0.426i)T \)
23 \( 1 + (0.376 - 0.926i)T \)
29 \( 1 + (-0.981 + 0.191i)T \)
31 \( 1 + (0.272 - 0.962i)T \)
37 \( 1 + (0.298 - 0.954i)T \)
41 \( 1 + (0.892 - 0.451i)T \)
43 \( 1 + (0.677 + 0.735i)T \)
47 \( 1 + (0.0550 + 0.998i)T \)
53 \( 1 + (0.879 + 0.475i)T \)
59 \( 1 + (-0.954 + 0.298i)T \)
61 \( 1 + (-0.546 - 0.837i)T \)
67 \( 1 + (0.892 + 0.451i)T \)
71 \( 1 + (-0.716 - 0.697i)T \)
73 \( 1 + (-0.936 + 0.350i)T \)
79 \( 1 + (0.981 + 0.191i)T \)
83 \( 1 + (-0.975 - 0.218i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.635 - 0.771i)T \)
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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

−19.89103698684228620035456907760, −19.11528331822834160982212736803, −18.35138135445623398853333544238, −17.47258525950810121030085497499, −16.626321728360658029499193248160, −16.140675430170517365405949501217, −15.34231328758948576372996335403, −14.725637579369473382430219033198, −13.954956005878962474526558034488, −13.40328416487034710041741361626, −11.92916094285460716786951391027, −11.31399319032055196383272167792, −10.881471265211506157438708129754, −10.255070858561904109009165466328, −9.12300814983634434466103727287, −8.435068408395595175268907421522, −7.73865826568412483484463558470, −6.77742728907276735743843251804, −5.85472355817805938551214191408, −4.93847555592120926676396462284, −3.9831899017603853735828542748, −3.62029096585656560033675583952, −2.711904276372145038810998356174, −1.18031617045082508806912053928, −0.15400720230900459232548862590, 1.01618134212760808440082814327, 1.8225556734833349632452338909, 2.614522231849501536646329549820, 3.98103950060079210582942759184, 4.582150523581933138548028976427, 5.72629222988977329943508899991, 6.333493905540985848164594582740, 7.4066994638666904435722307879, 7.970270287276808145621897642573, 8.75905292085744155590470072399, 9.17749922493870515637050473626, 10.88297567923261127070617353509, 11.194759748505051871132453738685, 12.189100564600738270861081522890, 12.66441240582943651279173638442, 13.158790604005765345764968117873, 14.41397642203486790303986585818, 14.94269896960135105319744037316, 15.61465262698791229633604933030, 16.662729526307503462152206828513, 17.3318214379269584733755995191, 17.99873531443308953153703195157, 18.82056513585753103757973343499, 19.25898928005366547440346266469, 20.15167555176078054479351466832

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