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Properties

Label 2-20e2-16.13-c1-0-13
Degree $2$
Conductor $400$
Sign $0.880 - 0.474i$
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  + (−0.507 + 1.31i)2-s + (0.0623 + 0.0623i)3-s + (−1.48 − 1.34i)4-s + (−0.113 + 0.0506i)6-s + 0.375i·7-s + (2.52 − 1.27i)8-s − 2.99i·9-s + (2.36 − 2.36i)11-s + (−0.00895 − 0.176i)12-s + (1.76 + 1.76i)13-s + (−0.496 − 0.190i)14-s + (0.405 + 3.97i)16-s + 4.64·17-s + (3.94 + 1.51i)18-s + (−2.34 − 2.34i)19-s + ⋯
L(s)  = 1  + (−0.359 + 0.933i)2-s + (0.0359 + 0.0359i)3-s + (−0.742 − 0.670i)4-s + (−0.0465 + 0.0206i)6-s + 0.142i·7-s + (0.892 − 0.451i)8-s − 0.997i·9-s + (0.713 − 0.713i)11-s + (−0.00258 − 0.0508i)12-s + (0.489 + 0.489i)13-s + (−0.132 − 0.0510i)14-s + (0.101 + 0.994i)16-s + 1.12·17-s + (0.930 + 0.358i)18-s + (−0.539 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.880 - 0.474i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11003 + 0.280087i\)
\(L(\frac12)\) \(\approx\) \(1.11003 + 0.280087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.507 - 1.31i)T \)
5 \( 1 \)
good3 \( 1 + (-0.0623 - 0.0623i)T + 3iT^{2} \)
7 \( 1 - 0.375iT - 7T^{2} \)
11 \( 1 + (-2.36 + 2.36i)T - 11iT^{2} \)
13 \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \)
17 \( 1 - 4.64T + 17T^{2} \)
19 \( 1 + (2.34 + 2.34i)T + 19iT^{2} \)
23 \( 1 - 2.07iT - 23T^{2} \)
29 \( 1 + (-2.55 - 2.55i)T + 29iT^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 + (-7.62 + 7.62i)T - 37iT^{2} \)
41 \( 1 + 3.77iT - 41T^{2} \)
43 \( 1 + (6.21 - 6.21i)T - 43iT^{2} \)
47 \( 1 + 9.71T + 47T^{2} \)
53 \( 1 + (3.03 - 3.03i)T - 53iT^{2} \)
59 \( 1 + (8.11 - 8.11i)T - 59iT^{2} \)
61 \( 1 + (-0.728 - 0.728i)T + 61iT^{2} \)
67 \( 1 + (-0.969 - 0.969i)T + 67iT^{2} \)
71 \( 1 + 9.14iT - 71T^{2} \)
73 \( 1 + 7.56iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 + 15.7iT - 89T^{2} \)
97 \( 1 - 3.86T + 97T^{2} \)
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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

−11.30599687460756459504484206312, −10.16266196412835680417758917725, −9.242346951127547483724488307041, −8.689200175748967382709998249396, −7.63260006607461885872730575374, −6.44990565800646175470039676011, −6.00592474760232023985122136244, −4.58439725080458577956785907151, −3.43095856593442772349631385634, −1.06811647502963565388532643658, 1.39459954823878975391345142607, 2.78255673275285316057354356820, 4.08591190313718508647380656497, 5.06531889109715739036532482541, 6.55914490040958300057180044903, 7.976623070947362889521297529528, 8.303447005859016092304509357685, 9.793604981479769826191705344432, 10.15577188524127426340065521358, 11.18010343031680024103273679529

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