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Properties

Label 1-417-417.11-r1-0-0
Degree $1$
Conductor $417$
Sign $-0.747 + 0.664i$
Analytic cond. $44.8128$
Root an. cond. $44.8128$
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  + (0.949 + 0.313i)2-s + (0.803 + 0.595i)4-s + (0.648 − 0.761i)5-s + (−0.974 − 0.225i)7-s + (0.576 + 0.816i)8-s + (0.854 − 0.519i)10-s + (−0.613 + 0.789i)11-s + (0.0227 + 0.999i)13-s + (−0.854 − 0.519i)14-s + (0.291 + 0.956i)16-s + (−0.898 + 0.439i)17-s + (−0.829 − 0.557i)19-s + (0.974 − 0.225i)20-s + (−0.829 + 0.557i)22-s + (−0.682 + 0.730i)23-s + ⋯
L(s)  = 1  + (0.949 + 0.313i)2-s + (0.803 + 0.595i)4-s + (0.648 − 0.761i)5-s + (−0.974 − 0.225i)7-s + (0.576 + 0.816i)8-s + (0.854 − 0.519i)10-s + (−0.613 + 0.789i)11-s + (0.0227 + 0.999i)13-s + (−0.854 − 0.519i)14-s + (0.291 + 0.956i)16-s + (−0.898 + 0.439i)17-s + (−0.829 − 0.557i)19-s + (0.974 − 0.225i)20-s + (−0.829 + 0.557i)22-s + (−0.682 + 0.730i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(44.8128\)
Root analytic conductor: \(44.8128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (1:\ ),\ -0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7428719735 + 1.954119336i\)
\(L(\frac12)\) \(\approx\) \(0.7428719735 + 1.954119336i\)
\(L(1)\) \(\approx\) \(1.476066187 + 0.5236566256i\)
\(L(1)\) \(\approx\) \(1.476066187 + 0.5236566256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.949 + 0.313i)T \)
5 \( 1 + (0.648 - 0.761i)T \)
7 \( 1 + (-0.974 - 0.225i)T \)
11 \( 1 + (-0.613 + 0.789i)T \)
13 \( 1 + (0.0227 + 0.999i)T \)
17 \( 1 + (-0.898 + 0.439i)T \)
19 \( 1 + (-0.829 - 0.557i)T \)
23 \( 1 + (-0.682 + 0.730i)T \)
29 \( 1 + (-0.113 + 0.993i)T \)
31 \( 1 + (0.538 - 0.842i)T \)
37 \( 1 + (0.934 + 0.356i)T \)
41 \( 1 + (0.715 + 0.699i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.983 + 0.181i)T \)
53 \( 1 + (-0.746 + 0.665i)T \)
59 \( 1 + (0.917 + 0.398i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (0.377 - 0.926i)T \)
71 \( 1 + (-0.983 - 0.181i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (0.962 + 0.269i)T \)
83 \( 1 + (-0.377 - 0.926i)T \)
89 \( 1 + (0.158 - 0.987i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−23.39468822635867145689497057674, −22.66009560258738549506094290835, −22.10297163194590230750813561911, −21.308106324307581641512822421547, −20.432571009845587526923177231878, −19.379878613230644498175241383979, −18.722778598274320102509161651660, −17.73194148169934645665147251992, −16.38185791513294809864593981892, −15.63895320397643029745613962956, −14.807818902168081078755330825604, −13.77842623161688105066703434581, −13.17513953009612731236992307732, −12.37724707500408740239715541342, −11.09400086987734819335718824686, −10.424275805878665875332769136384, −9.66300302218663534568444398343, −8.16277209131867822008609089746, −6.75504354431462429178513243200, −6.138443230917824811295846209, −5.34064067291584369777719508522, −3.873787556266354235869393331016, −2.876271140932623267176769012934, −2.24200613125523526004550494028, −0.34860009990256245946339603505, 1.72652588970226694127442481328, 2.69339541994581518080480777495, 4.16823342366576384237209082601, 4.77262560531375940320785177219, 6.08097563764808312721762087565, 6.63776966588381154355111986431, 7.836067875447288062760747490515, 9.0602424499214174409037901087, 9.96216617267649583671108878642, 11.145542851234507304432622982548, 12.29528327283288297015505592925, 13.12163243190075886537262051848, 13.42895194558767781753193410052, 14.65241212943479911750484654075, 15.62768093424842158752500617563, 16.383043375742235200235990320469, 17.103637741618144956331826203324, 18.03841541707960817072869597165, 19.588193595200224969575732533186, 20.1120131570752786884007127494, 21.19199925043350381510229185136, 21.73676059841949475263980900558, 22.6480208996034289213190486366, 23.66862184342268586211035358107, 24.0800273837357591194317098593

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