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Properties

Label 1-1832-1832.819-r1-0-0
Degree $1$
Conductor $1832$
Sign $0.408 + 0.912i$
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  + (0.635 − 0.771i)3-s + (−0.851 − 0.523i)5-s + (0.592 − 0.805i)7-s + (−0.191 − 0.981i)9-s + (−0.0825 − 0.996i)11-s + (0.879 − 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.879 + 0.475i)17-s + (0.0275 + 0.999i)19-s + (−0.245 − 0.969i)21-s + (−0.975 + 0.218i)23-s + (0.451 + 0.892i)25-s + (−0.879 − 0.475i)27-s + (−0.993 − 0.110i)29-s + (0.821 − 0.569i)31-s + ⋯
L(s)  = 1  + (0.635 − 0.771i)3-s + (−0.851 − 0.523i)5-s + (0.592 − 0.805i)7-s + (−0.191 − 0.981i)9-s + (−0.0825 − 0.996i)11-s + (0.879 − 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.879 + 0.475i)17-s + (0.0275 + 0.999i)19-s + (−0.245 − 0.969i)21-s + (−0.975 + 0.218i)23-s + (0.451 + 0.892i)25-s + (−0.879 − 0.475i)27-s + (−0.993 − 0.110i)29-s + (0.821 − 0.569i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ 0.408 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1154333439 + 0.07477894592i\)
\(L(\frac12)\) \(\approx\) \(0.1154333439 + 0.07477894592i\)
\(L(1)\) \(\approx\) \(0.8835598514 - 0.5018278905i\)
\(L(1)\) \(\approx\) \(0.8835598514 - 0.5018278905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.635 - 0.771i)T \)
5 \( 1 + (-0.851 - 0.523i)T \)
7 \( 1 + (0.592 - 0.805i)T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (0.879 - 0.475i)T \)
17 \( 1 + (-0.879 + 0.475i)T \)
19 \( 1 + (0.0275 + 0.999i)T \)
23 \( 1 + (-0.975 + 0.218i)T \)
29 \( 1 + (-0.993 - 0.110i)T \)
31 \( 1 + (0.821 - 0.569i)T \)
37 \( 1 + (0.962 + 0.272i)T \)
41 \( 1 + (-0.754 - 0.656i)T \)
43 \( 1 + (0.245 + 0.969i)T \)
47 \( 1 + (0.191 + 0.981i)T \)
53 \( 1 + (0.986 + 0.164i)T \)
59 \( 1 + (-0.962 + 0.272i)T \)
61 \( 1 + (-0.945 - 0.324i)T \)
67 \( 1 + (-0.754 + 0.656i)T \)
71 \( 1 + (-0.904 + 0.426i)T \)
73 \( 1 + (0.451 + 0.892i)T \)
79 \( 1 + (-0.993 + 0.110i)T \)
83 \( 1 + (0.716 + 0.697i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.998 + 0.0550i)T \)
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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

−20.05903556372269688340282082089, −19.18906722222553904536680165173, −18.30203295985770895427548582518, −17.94086072170295381479232514488, −16.67440930158785193577443141872, −15.87170274241028694016324443513, −15.250002552716063490459331999426, −15.03050895593014684498819480034, −14.032382592987720317930810104403, −13.37225665723528841803351926179, −12.18310504634615939143667421788, −11.4990972289507540092657538363, −10.91405876677512405531010663650, −10.074062470475043141354151457290, −9.057676429837684509677579173556, −8.64826507629705972427977596049, −7.749058678077818960210505101360, −7.03257580297421872646616202147, −5.97584418963941157597173004195, −4.71905827985044404827333952922, −4.43152909096674537355513145047, −3.41524111560304398095730904694, −2.51148436230349815299812917753, −1.83451854877559120881950370738, −0.02361797959498890887000963488, 0.97264943995061337163084320606, 1.57722628775628500666426015512, 2.87842267878812587209542240992, 3.87669799142033202072512032742, 4.18488284309541092339652075722, 5.662233680087091395933576657839, 6.33351183304881351348960966602, 7.46388053081555550604257350156, 8.08566095030952125228400331119, 8.36596264299910229935035063090, 9.313787231685235224117648510464, 10.509581169133950710718026493455, 11.27807935652943601243589247518, 11.87093071212305675466288604703, 12.86600759587620868198371524114, 13.423739380154984014035120123275, 14.02239169047451512738900211861, 14.92162834396701589284825874079, 15.609574557372783734514654776667, 16.46936227805215169662601201956, 17.16799865023484497493108738932, 18.07739276258589885877406972983, 18.70088435541456614938697622965, 19.442490008658852882885829430119, 20.12721330542747161824015876560

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