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Properties

Label 2-40e2-80.43-c1-0-16
Degree $2$
Conductor $1600$
Sign $0.663 + 0.748i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·3-s + (−0.896 + 0.896i)7-s − 0.732·9-s + (−4.09 + 4.09i)11-s + 4.89·13-s + (0.707 − 0.707i)17-s + (3.09 − 3.09i)19-s + (1.73 + 1.73i)21-s + (2.96 + 2.96i)23-s − 4.38i·27-s + (−1.26 − 1.26i)29-s − 4.19i·31-s + (7.91 + 7.91i)33-s + 10.9·37-s − 9.46i·39-s + ⋯
L(s)  = 1  − 1.11i·3-s + (−0.338 + 0.338i)7-s − 0.244·9-s + (−1.23 + 1.23i)11-s + 1.35·13-s + (0.171 − 0.171i)17-s + (0.710 − 0.710i)19-s + (0.377 + 0.377i)21-s + (0.618 + 0.618i)23-s − 0.843i·27-s + (−0.235 − 0.235i)29-s − 0.753i·31-s + (1.37 + 1.37i)33-s + 1.79·37-s − 1.51i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695796107\)
\(L(\frac12)\) \(\approx\) \(1.695796107\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93iT - 3T^{2} \)
7 \( 1 + (0.896 - 0.896i)T - 7iT^{2} \)
11 \( 1 + (4.09 - 4.09i)T - 11iT^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - 17iT^{2} \)
19 \( 1 + (-3.09 + 3.09i)T - 19iT^{2} \)
23 \( 1 + (-2.96 - 2.96i)T + 23iT^{2} \)
29 \( 1 + (1.26 + 1.26i)T + 29iT^{2} \)
31 \( 1 + 4.19iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 - 9.14T + 43T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + 9.89iT - 53T^{2} \)
59 \( 1 + (4.26 + 4.26i)T + 59iT^{2} \)
61 \( 1 + (-7.19 + 7.19i)T - 61iT^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (3.91 - 3.91i)T - 73iT^{2} \)
79 \( 1 + 8.19T + 79T^{2} \)
83 \( 1 - 4.65iT - 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + (-3.10 + 3.10i)T - 97iT^{2} \)
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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

−9.437372253162058734208709757611, −8.207353695722480851527451296498, −7.68347468899340867477072819271, −6.98791225144160673153661937194, −6.17446680970462436184876085476, −5.34907865271949389768821060935, −4.30387841721185075413116933118, −2.97414894909233668343380497104, −2.11549448888050874483016525523, −0.909296249117543762294682115797, 0.979269419352819304230081351144, 2.87987500721494508316613651643, 3.58267999649906873905744379826, 4.37302797780677141911420400779, 5.50200159900737298469611182318, 5.95307743723032000882065106481, 7.17554077559611571738535436874, 8.112364019807804515669559984800, 8.796360414275972619838033199074, 9.551746701282137461082448273725

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