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Properties

Label 2-40e2-20.19-c2-0-40
Degree $2$
Conductor $1600$
Sign $0.894 - 0.447i$
Analytic cond. $43.5968$
Root an. cond. $6.60279$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.23·3-s + 3.23·7-s + 1.47·9-s − 15.4i·11-s + 21.4i·13-s + 10.9i·17-s + 5.88i·19-s + 10.4·21-s + 21.1·23-s − 24.3·27-s + 38.9·29-s + 25.5i·31-s − 49.8i·33-s + 43.3i·37-s + 69.3i·39-s + ⋯
L(s)  = 1  + 1.07·3-s + 0.462·7-s + 0.163·9-s − 1.40i·11-s + 1.64i·13-s + 0.643i·17-s + 0.309i·19-s + 0.498·21-s + 0.918·23-s − 0.902·27-s + 1.34·29-s + 0.823i·31-s − 1.51i·33-s + 1.17i·37-s + 1.77i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.129040173\)
\(L(\frac12)\) \(\approx\) \(3.129040173\)
\(L(2)\) 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 \( 1 - 3.23T + 9T^{2} \)
7 \( 1 - 3.23T + 49T^{2} \)
11 \( 1 + 15.4iT - 121T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 - 10.9iT - 289T^{2} \)
19 \( 1 - 5.88iT - 361T^{2} \)
23 \( 1 - 21.1T + 529T^{2} \)
29 \( 1 - 38.9T + 841T^{2} \)
31 \( 1 - 25.5iT - 961T^{2} \)
37 \( 1 - 43.3iT - 1.36e3T^{2} \)
41 \( 1 - 58.1T + 1.68e3T^{2} \)
43 \( 1 - 67.2T + 1.84e3T^{2} \)
47 \( 1 + 21.4T + 2.20e3T^{2} \)
53 \( 1 + 18.1iT - 2.80e3T^{2} \)
59 \( 1 + 108. iT - 3.48e3T^{2} \)
61 \( 1 - 97.1T + 3.72e3T^{2} \)
67 \( 1 - 71.0T + 4.48e3T^{2} \)
71 \( 1 - 39.6iT - 5.04e3T^{2} \)
73 \( 1 - 18.7iT - 5.32e3T^{2} \)
79 \( 1 + 36.4iT - 6.24e3T^{2} \)
83 \( 1 - 129.T + 6.88e3T^{2} \)
89 \( 1 + 117.T + 7.92e3T^{2} \)
97 \( 1 - 81.3iT - 9.40e3T^{2} \)
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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

−9.085084641676391780670598011844, −8.434653183009729116766503576193, −8.071870612781991404290390015211, −6.87857891861273168022704414477, −6.17596467947750682829737551897, −5.05256230105823585547239443116, −4.04520142215695482647238922828, −3.23013333642643740191717952358, −2.29340178446963732718574709849, −1.14764455615280833800060954031, 0.840989805874606887043687779911, 2.35536756018503853126647702235, 2.84413013908988596046606506896, 4.04916871124809959909575697417, 4.93876552462177540608179008541, 5.79472199429369414770693612478, 7.11647948688716385785225468695, 7.66408246632217576099198164625, 8.296184465133809743002141964534, 9.182815985699355692782590397068

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