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Properties

Label 2-20e2-16.13-c1-0-21
Degree $2$
Conductor $400$
Sign $0.659 - 0.751i$
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  + (1.35 + 0.388i)2-s + (1.03 + 1.03i)3-s + (1.69 + 1.05i)4-s + (1.01 + 1.81i)6-s − 1.49i·7-s + (1.89 + 2.09i)8-s − 0.836i·9-s + (0.423 − 0.423i)11-s + (0.666 + 2.86i)12-s + (1.85 + 1.85i)13-s + (0.581 − 2.03i)14-s + (1.76 + 3.58i)16-s − 6.50·17-s + (0.325 − 1.13i)18-s + (−1.75 − 1.75i)19-s + ⋯
L(s)  = 1  + (0.961 + 0.274i)2-s + (0.600 + 0.600i)3-s + (0.849 + 0.528i)4-s + (0.412 + 0.742i)6-s − 0.565i·7-s + (0.671 + 0.741i)8-s − 0.278i·9-s + (0.127 − 0.127i)11-s + (0.192 + 0.827i)12-s + (0.515 + 0.515i)13-s + (0.155 − 0.543i)14-s + (0.441 + 0.897i)16-s − 1.57·17-s + (0.0766 − 0.268i)18-s + (−0.403 − 0.403i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.659 - 0.751i$
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: \((2,\ 400,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58960 + 1.17230i\)
\(L(\frac12)\) \(\approx\) \(2.58960 + 1.17230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.388i)T \)
5 \( 1 \)
good3 \( 1 + (-1.03 - 1.03i)T + 3iT^{2} \)
7 \( 1 + 1.49iT - 7T^{2} \)
11 \( 1 + (-0.423 + 0.423i)T - 11iT^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 + 6.50T + 17T^{2} \)
19 \( 1 + (1.75 + 1.75i)T + 19iT^{2} \)
23 \( 1 - 7.19iT - 23T^{2} \)
29 \( 1 + (6.57 + 6.57i)T + 29iT^{2} \)
31 \( 1 + 6.75T + 31T^{2} \)
37 \( 1 + (1.95 - 1.95i)T - 37iT^{2} \)
41 \( 1 + 7.70iT - 41T^{2} \)
43 \( 1 + (-6.13 + 6.13i)T - 43iT^{2} \)
47 \( 1 - 6.65T + 47T^{2} \)
53 \( 1 + (5.29 - 5.29i)T - 53iT^{2} \)
59 \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \)
61 \( 1 + (1.43 + 1.43i)T + 61iT^{2} \)
67 \( 1 + (-6.35 - 6.35i)T + 67iT^{2} \)
71 \( 1 - 4.08iT - 71T^{2} \)
73 \( 1 + 2.43iT - 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (2.81 + 2.81i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−11.31882962091781004333763105572, −10.79028033855448701381557233958, −9.416198611734468631687995042234, −8.724395189561959264800272486080, −7.46270097848128472412369246589, −6.63831360199254143802552145738, −5.52902822008514472315183007253, −4.08487245027669400888291316243, −3.80020830423017680902026643815, −2.22126381900086882861725382774, 1.84707809157056461338657992796, 2.75729115148102497889878853465, 4.10708093743519607881019862792, 5.27732677660795652798243653522, 6.35241833578953384390376307886, 7.23453188359593186353590364411, 8.342899990056513254321613486085, 9.213104358878180774238401926695, 10.71933953706985857698845953814, 11.07470874969153551229728955904

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