Properties

Label 2-20e2-5.4-c1-0-2
Degree $2$
Conductor $400$
Sign $0.447 - 0.894i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·7-s + 3·9-s − 4·11-s + 2i·13-s + 2i·17-s + 4·19-s + 4i·23-s + 2·29-s + 8·31-s + 6i·37-s − 6·41-s − 8i·43-s − 4i·47-s − 9·49-s − 6i·53-s + ⋯
L(s)  = 1  + 1.51i·7-s + 9-s − 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 0.917·19-s + 0.834i·23-s + 0.371·29-s + 1.43·31-s + 0.986i·37-s − 0.937·41-s − 1.21i·43-s − 0.583i·47-s − 1.28·49-s − 0.824i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12940 + 0.698012i\)
\(L(\frac12)\) \(\approx\) \(1.12940 + 0.698012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66637468486795400945844347698, −10.36371787716092953955720282987, −9.687941945913011033985574344810, −8.669159741380960439502819043824, −7.82345485548024244995230582501, −6.69750607834347011257043200244, −5.57782202616065436043968572110, −4.74860444575389548886244759317, −3.16295329624880242409614353354, −1.89590827212659631097713103959, 0.939278969634963650164151672092, 2.89662801195393895582887534117, 4.22963467981581805606017116424, 5.08020078559631303189439793818, 6.56463226477988062593447218137, 7.49266168316699170013362263022, 8.019356680266294677456073115161, 9.569900651218649872082119047581, 10.34432295746027101544622215853, 10.77212921437032853485016691296

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