Properties

Label 2-800-5.4-c5-0-15
Degree $2$
Conductor $800$
Sign $-0.894 - 0.447i$
Analytic cond. $128.307$
Root an. cond. $11.3272$
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  − 0.755i·3-s + 172. i·7-s + 242.·9-s − 391.·11-s − 149. i·13-s + 1.18e3i·17-s + 685.·19-s + 130.·21-s + 996. i·23-s − 366. i·27-s + 8.76e3·29-s − 9.52e3·31-s + 295. i·33-s + 1.02e4i·37-s − 112.·39-s + ⋯
L(s)  = 1  − 0.0484i·3-s + 1.33i·7-s + 0.997·9-s − 0.975·11-s − 0.244i·13-s + 0.996i·17-s + 0.435·19-s + 0.0644·21-s + 0.392i·23-s − 0.0968i·27-s + 1.93·29-s − 1.78·31-s + 0.0472i·33-s + 1.22i·37-s − 0.0118·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(128.307\)
Root analytic conductor: \(11.3272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :5/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.315595240\)
\(L(\frac12)\) \(\approx\) \(1.315595240\)
\(L(\frac{7}{2})\) 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 \( 1 + 0.755iT - 243T^{2} \)
7 \( 1 - 172. iT - 1.68e4T^{2} \)
11 \( 1 + 391.T + 1.61e5T^{2} \)
13 \( 1 + 149. iT - 3.71e5T^{2} \)
17 \( 1 - 1.18e3iT - 1.41e6T^{2} \)
19 \( 1 - 685.T + 2.47e6T^{2} \)
23 \( 1 - 996. iT - 6.43e6T^{2} \)
29 \( 1 - 8.76e3T + 2.05e7T^{2} \)
31 \( 1 + 9.52e3T + 2.86e7T^{2} \)
37 \( 1 - 1.02e4iT - 6.93e7T^{2} \)
41 \( 1 - 32.6T + 1.15e8T^{2} \)
43 \( 1 - 1.03e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.69e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.22e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.42e4T + 7.14e8T^{2} \)
61 \( 1 + 2.17e4T + 8.44e8T^{2} \)
67 \( 1 + 2.07e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.53e4T + 1.80e9T^{2} \)
73 \( 1 + 5.79e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.24e4T + 3.07e9T^{2} \)
83 \( 1 + 4.35e4iT - 3.93e9T^{2} \)
89 \( 1 + 6.62e4T + 5.58e9T^{2} \)
97 \( 1 + 1.22e4iT - 8.58e9T^{2} \)
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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

−9.968532695775845427632545645869, −8.957627124361555033148118428810, −8.220178361743954247709723867857, −7.38673179743830816769745115859, −6.30929240576507647884570538603, −5.47312472797462897962350718093, −4.65554303200799932143016424426, −3.35198040128641101194588894883, −2.35116959640712297916446654181, −1.32286767428126394623184451968, 0.27681183389687099342744382260, 1.21357507274859323491321590128, 2.55442155886327519466000580246, 3.78141817063589530651375721245, 4.56692722171304283951590048245, 5.46150561942609743568482034057, 6.95802845569915738881736842882, 7.20900424848506809401022529105, 8.154080101409346909197454274523, 9.311519342706840130389196881269

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