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Properties

Label 1-20e2-400.163-r0-0-0
Degree $1$
Conductor $400$
Sign $-0.780 + 0.625i$
Analytic cond. $1.85759$
Root an. cond. $1.85759$
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  + (−0.809 + 0.587i)3-s − i·7-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−0.309 + 0.951i)13-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)19-s + (0.587 + 0.809i)21-s + (−0.951 + 0.309i)23-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)3-s − i·7-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−0.309 + 0.951i)13-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)19-s + (0.587 + 0.809i)21-s + (−0.951 + 0.309i)23-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(1.85759\)
Root analytic conductor: \(1.85759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (0:\ ),\ -0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1361366901 + 0.3876605351i\)
\(L(\frac12)\) \(\approx\) \(0.1361366901 + 0.3876605351i\)
\(L(1)\) \(\approx\) \(0.6182584873 + 0.1588678174i\)
\(L(1)\) \(\approx\) \(0.6182584873 + 0.1588678174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−24.28246618916523510760160366728, −22.95807458352101776818197246393, −22.53783957252884157301762065807, −21.598128580299261312136888143840, −20.64638313306267980970407361878, −19.50849662342221793682060869695, −18.479752500674838801811055360067, −18.13477357573103748280546518222, −17.19366214537328049240175689356, −15.93763944177045399014237555274, −15.59593403640639191501174605065, −14.13026200053435489231345710019, −13.19121617801598011941641921787, −12.32467516278085997151267689551, −11.68282997268812460666525049617, −10.61817581394707898206445965017, −9.72372547786250085694270842604, −8.26463209896971486001690569609, −7.65639750771766760993762249074, −6.30163783896814579888458786961, −5.57927271547639402814615044344, −4.76405917149432609078017127825, −2.95435456911883077457593818685, −2.00087878739302388283604247619, −0.26498896849805662510875437267, 1.4849072559701040194802770840, 3.19423345920826780639846317626, 4.42379567512762485348245704886, 4.97063282181268719199447103415, 6.381547009304692897532133789520, 7.09638501580768999928069440205, 8.36050226928974737188661271015, 9.696352261582907299493026715903, 10.317745644295511603909274370014, 11.17978031956272659855497629827, 12.06621163534045202674342708488, 13.16458028899732984219976182003, 14.05965441997778211690833184554, 15.255237787850840062323023212681, 15.99529340806297218743183595387, 16.84315920656300787707281475505, 17.6003097099340392480132496349, 18.361142971987097318297789200312, 19.71576913514370114108269537847, 20.394163233916193663641125164545, 21.503569060429191687406317080926, 21.95362593893559641309110104648, 23.191831653544505384439653228531, 23.63542472818526135622546664668, 24.3720672533461280870528358469

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