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Properties

Label 2-20e2-20.7-c3-0-6
Degree $2$
Conductor $400$
Sign $-0.473 - 0.880i$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.12 + 4.12i)3-s + (4.12 − 4.12i)7-s + 7i·9-s + 13.8i·11-s + (−57.1 + 57.1i)13-s + (57.1 + 57.1i)17-s − 96.9·19-s + 34·21-s + (86.5 + 86.5i)23-s + (82.4 − 82.4i)27-s + 174i·29-s + 193. i·31-s + (−57.1 + 57.1i)33-s − 471.·39-s + 252·41-s + ⋯
L(s)  = 1  + (0.793 + 0.793i)3-s + (0.222 − 0.222i)7-s + 0.259i·9-s + 0.379i·11-s + (−1.21 + 1.21i)13-s + (0.815 + 0.815i)17-s − 1.17·19-s + 0.353·21-s + (0.784 + 0.784i)23-s + (0.587 − 0.587i)27-s + 1.11i·29-s + 1.12i·31-s + (−0.301 + 0.301i)33-s − 1.93·39-s + 0.959·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.011876959\)
\(L(\frac12)\) \(\approx\) \(2.011876959\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.12 - 4.12i)T + 27iT^{2} \)
7 \( 1 + (-4.12 + 4.12i)T - 343iT^{2} \)
11 \( 1 - 13.8iT - 1.33e3T^{2} \)
13 \( 1 + (57.1 - 57.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-57.1 - 57.1i)T + 4.91e3iT^{2} \)
19 \( 1 + 96.9T + 6.85e3T^{2} \)
23 \( 1 + (-86.5 - 86.5i)T + 1.21e4iT^{2} \)
29 \( 1 - 174iT - 2.43e4T^{2} \)
31 \( 1 - 193. iT - 2.97e4T^{2} \)
37 \( 1 + 5.06e4iT^{2} \)
41 \( 1 - 252T + 6.89e4T^{2} \)
43 \( 1 + (202. + 202. i)T + 7.95e4iT^{2} \)
47 \( 1 + (284. - 284. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-399. + 399. i)T - 1.48e5iT^{2} \)
59 \( 1 + 872.T + 2.05e5T^{2} \)
61 \( 1 - 56T + 2.26e5T^{2} \)
67 \( 1 + (-317. + 317. i)T - 3.00e5iT^{2} \)
71 \( 1 - 387. iT - 3.57e5T^{2} \)
73 \( 1 + (-399. + 399. i)T - 3.89e5iT^{2} \)
79 \( 1 + 692.T + 4.93e5T^{2} \)
83 \( 1 + (482. + 482. i)T + 5.71e5iT^{2} \)
89 \( 1 + 42iT - 7.04e5T^{2} \)
97 \( 1 + (-742. - 742. i)T + 9.12e5iT^{2} \)
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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

−10.93017239685338709845580532278, −10.09716630190112856542369700739, −9.327737016116147635445110583765, −8.623966672456710142386857002356, −7.50851651319262023753276800550, −6.55914083197890199646097154248, −5.01068367853616940845056388115, −4.19654472464386216786284809452, −3.12534577402235047806552712648, −1.73597164416406837914943966650, 0.59067455016078825674050061154, 2.25810725920020901578106912903, 3.00204570679180216492638140868, 4.67022946941368958306638954485, 5.77137033998383246561767049476, 7.04423234744765986746378663879, 7.86430388185372811086093460853, 8.432832189953926080302252958789, 9.572438848809819570642774433672, 10.48401229858930998345567404021

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