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Properties

Label 2-20e2-20.7-c1-0-5
Degree $2$
Conductor $400$
Sign $0.997 + 0.0706i$
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  + (0.707 + 0.707i)3-s + (2.82 − 2.82i)7-s − 1.99i·9-s + 5.19i·11-s + (2.44 − 2.44i)13-s + (−3.67 − 3.67i)17-s + 1.73·19-s + 4.00·21-s + (4.24 + 4.24i)23-s + (3.53 − 3.53i)27-s + 6i·29-s + 3.46i·31-s + (−3.67 + 3.67i)33-s + 3.46·39-s − 3·41-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.06 − 1.06i)7-s − 0.666i·9-s + 1.56i·11-s + (0.679 − 0.679i)13-s + (−0.891 − 0.891i)17-s + 0.397·19-s + 0.872·21-s + (0.884 + 0.884i)23-s + (0.680 − 0.680i)27-s + 1.11i·29-s + 0.622i·31-s + (−0.639 + 0.639i)33-s + 0.554·39-s − 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.997 + 0.0706i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.997 + 0.0706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73435 - 0.0613380i\)
\(L(\frac12)\) \(\approx\) \(1.73435 - 0.0613380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T + 3iT^{2} \)
7 \( 1 + (-2.82 + 2.82i)T - 7iT^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 13iT^{2} \)
17 \( 1 + (3.67 + 3.67i)T + 17iT^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + (4.24 - 4.24i)T - 47iT^{2} \)
53 \( 1 + (-7.34 + 7.34i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + (4.94 - 4.94i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-2.12 - 2.12i)T + 83iT^{2} \)
89 \( 1 + 3iT - 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−11.14080355069659996737571419743, −10.34352429035311339464725630906, −9.467578754569255817489847642085, −8.594643653852961247472562916229, −7.44384324506720943608430656900, −6.86274027469740014212543092107, −5.12189861825020120328053398149, −4.37026076728613397483050326301, −3.21717340930650350162973038738, −1.41583616916667979475260290415, 1.68422180276530823810069757303, 2.82864408971800276007456533623, 4.41014494758510023773611720287, 5.58089464276196151576911537406, 6.45957689202116199393714209385, 7.902365388996383775229830180709, 8.522415283291677863971993934202, 9.012424464442882103606496211911, 10.71232060666922649038755502695, 11.23956158728093874440363037894

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