Properties

Label 2-20e2-20.3-c5-0-11
Degree $2$
Conductor $400$
Sign $-0.880 - 0.473i$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.48 + 7.48i)3-s + (−19.2 − 19.2i)7-s + 131. i·9-s + 180. i·11-s + (−44.2 − 44.2i)13-s + (621. − 621. i)17-s + 2.67e3·19-s + 287.·21-s + (−2.23e3 + 2.23e3i)23-s + (−2.79e3 − 2.79e3i)27-s + 705. i·29-s + 2.76e3i·31-s + (−1.35e3 − 1.35e3i)33-s + (3.54e3 − 3.54e3i)37-s + 661.·39-s + ⋯
L(s)  = 1  + (−0.480 + 0.480i)3-s + (−0.148 − 0.148i)7-s + 0.539i·9-s + 0.450i·11-s + (−0.0725 − 0.0725i)13-s + (0.521 − 0.521i)17-s + 1.69·19-s + 0.142·21-s + (−0.880 + 0.880i)23-s + (−0.738 − 0.738i)27-s + 0.155i·29-s + 0.516i·31-s + (−0.216 − 0.216i)33-s + (0.425 − 0.425i)37-s + 0.0696·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.011398281\)
\(L(\frac12)\) \(\approx\) \(1.011398281\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.48 - 7.48i)T - 243iT^{2} \)
7 \( 1 + (19.2 + 19.2i)T + 1.68e4iT^{2} \)
11 \( 1 - 180. iT - 1.61e5T^{2} \)
13 \( 1 + (44.2 + 44.2i)T + 3.71e5iT^{2} \)
17 \( 1 + (-621. + 621. i)T - 1.41e6iT^{2} \)
19 \( 1 - 2.67e3T + 2.47e6T^{2} \)
23 \( 1 + (2.23e3 - 2.23e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 705. iT - 2.05e7T^{2} \)
31 \( 1 - 2.76e3iT - 2.86e7T^{2} \)
37 \( 1 + (-3.54e3 + 3.54e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.09e4T + 1.15e8T^{2} \)
43 \( 1 + (5.34e3 - 5.34e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.32e4 - 1.32e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (-1.56e4 - 1.56e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 4.59e4T + 7.14e8T^{2} \)
61 \( 1 + 1.75e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.10e4 - 3.10e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 1.06e4iT - 1.80e9T^{2} \)
73 \( 1 + (5.09e4 + 5.09e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 2.47e4T + 3.07e9T^{2} \)
83 \( 1 + (4.89e4 - 4.89e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 2.61e4iT - 5.58e9T^{2} \)
97 \( 1 + (3.99e4 - 3.99e4i)T - 8.58e9iT^{2} \)
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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

−10.80886795930712482151281598154, −9.912000166913618869777867104510, −9.344569284933051143793004713559, −7.85996110892331572630314053495, −7.27611414799444039223219269905, −5.83117496482327899179235543575, −5.13179362841029388652891381060, −4.05479769733606901618161779220, −2.78401035387803481303647170285, −1.25737456047239116134134240870, 0.29424390781487581980526424777, 1.35595837749519278103062343107, 2.90887951688020808865324002224, 4.05498370307076342701916566544, 5.52905893637239161903358073302, 6.15998508807161369010466532263, 7.20329068732234389829204412890, 8.120978540951432184813211593865, 9.240528242403332183067535349612, 10.04124444654467692656893639903

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