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Properties

Label 1-20e2-400.13-r1-0-0
Degree $1$
Conductor $400$
Sign $0.625 + 0.780i$
Analytic cond. $42.9859$
Root an. cond. $42.9859$
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

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

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ 0.625 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.251534831 + 1.081171626i\)
\(L(\frac12)\) \(\approx\) \(2.251534831 + 1.081171626i\)
\(L(1)\) \(\approx\) \(1.424557778 + 0.09762370708i\)
\(L(1)\) \(\approx\) \(1.424557778 + 0.09762370708i\)

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

−24.196790719343059211537535739876, −22.97290882671153087970788960717, −22.34263183564035206700797316163, −21.32671770676241672140052474682, −20.41718480721112675489091708517, −19.83138256247949809488404160062, −19.196867943539300269057355706677, −17.68099451685725147232803441256, −17.08239734147739260502051521943, −15.96759378849859421874012384462, −15.18150811791691618099586180699, −14.29633425817093497533289858210, −13.555182310885088203176106473931, −12.64837891661690064536107240139, −11.202544271252825938579409206424, −10.4695499002588237987823840837, −9.48054476663525500149465394869, −8.73966069840770294899683692557, −7.51311475105552697953823977317, −6.84341328402981452358251835275, −5.152332694587012069627951677028, −4.26097221845523997485027747258, −3.350377124284958566595840264446, −2.17468056956061965000760581784, −0.62129384838006560688190881823, 1.36851379177268183192979884846, 2.25961244973493453057325765420, 3.416844372024395292747439379370, 4.52747939469047356577997581585, 6.12625075859434827138640426287, 6.71240745300976633222070531939, 8.01291715502573008561053125514, 8.914211680105157167991569790338, 9.36802107181479112091731595989, 10.90451532268563641670838718330, 12.05118597916925822633807688905, 12.59967287610181080056132866935, 13.7081633704224505253281407516, 14.66072455992453597735493529427, 15.09200005761498201215631363615, 16.411009836730987766144166068625, 17.3502996626757321782659545032, 18.468494567355096287891314766805, 19.0950617421071004237331213143, 19.69607659742357092797038134784, 20.84804861080563064290555499509, 21.61841746571804470321513025199, 22.42399184857229175622347789924, 23.76869600719274042982631143987, 24.3212371353724449026069629999

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