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Properties

Label 2-20e2-5.2-c4-0-24
Degree $2$
Conductor $400$
Sign $0.850 + 0.525i$
Analytic cond. $41.3479$
Root an. cond. $6.43023$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9 − 9i)3-s + (29 + 29i)7-s − 81i·9-s + 118·11-s + (−69 + 69i)13-s + (271 + 271i)17-s − 280i·19-s + 522·21-s + (269 − 269i)23-s + 680i·29-s − 202·31-s + (1.06e3 − 1.06e3i)33-s + (651 + 651i)37-s + 1.24e3i·39-s + 1.68e3·41-s + ⋯
L(s)  = 1  + (1 − i)3-s + (0.591 + 0.591i)7-s − i·9-s + 0.975·11-s + (−0.408 + 0.408i)13-s + (0.937 + 0.937i)17-s − 0.775i·19-s + 1.18·21-s + (0.508 − 0.508i)23-s + 0.808i·29-s − 0.210·31-s + (0.975 − 0.975i)33-s + (0.475 + 0.475i)37-s + 0.816i·39-s + 1.00·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(41.3479\)
Root analytic conductor: \(6.43023\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :2),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.469299011\)
\(L(\frac12)\) \(\approx\) \(3.469299011\)
\(L(3)\) 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 \( 1 + (-9 + 9i)T - 81iT^{2} \)
7 \( 1 + (-29 - 29i)T + 2.40e3iT^{2} \)
11 \( 1 - 118T + 1.46e4T^{2} \)
13 \( 1 + (69 - 69i)T - 2.85e4iT^{2} \)
17 \( 1 + (-271 - 271i)T + 8.35e4iT^{2} \)
19 \( 1 + 280iT - 1.30e5T^{2} \)
23 \( 1 + (-269 + 269i)T - 2.79e5iT^{2} \)
29 \( 1 - 680iT - 7.07e5T^{2} \)
31 \( 1 + 202T + 9.23e5T^{2} \)
37 \( 1 + (-651 - 651i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.08e3 + 1.08e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.26e3 - 1.26e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-611 + 611i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.59e3T + 1.38e7T^{2} \)
67 \( 1 + (751 + 751i)T + 2.01e7iT^{2} \)
71 \( 1 + 6.44e3T + 2.54e7T^{2} \)
73 \( 1 + (-2.95e3 + 2.95e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 1.05e4iT - 3.89e7T^{2} \)
83 \( 1 + (6.23e3 - 6.23e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + (-7.31e3 - 7.31e3i)T + 8.85e7iT^{2} \)
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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

−10.61431770782391498920676357217, −9.214644585930375561708907635512, −8.758057545700998970755479694834, −7.79584679334781230671968869862, −7.02487302543674168628672378106, −5.98113514551885394701355450656, −4.59922336287925337853237962300, −3.21403600157058459026568554671, −2.10175949896515988789087562184, −1.16784728613533823293375304908, 1.09687482707370030536403764016, 2.72454516198388418838598037543, 3.79047159861015481394913400203, 4.53963381207783760655757274401, 5.76449284963179779151021792639, 7.33070941994476263076854480005, 7.991681759211568758212574433129, 9.130136390314551334347014665348, 9.672282478832618001564081507824, 10.51900781567236392780749389873

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