Properties

Label 2-40e2-40.13-c0-0-1
Degree $2$
Conductor $1600$
Sign $0.229 - 0.973i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·9-s + 2i·11-s − 2·19-s + 2·41-s + i·49-s + 2·59-s − 81-s − 2i·89-s − 2·99-s + ⋯
L(s)  = 1  + i·9-s + 2i·11-s − 2·19-s + 2·41-s + i·49-s + 2·59-s − 81-s − 2i·89-s − 2·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9772025443\)
\(L(\frac12)\) \(\approx\) \(0.9772025443\)
\(L(1)\) 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 \( 1 - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - 2iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.852823589809812008632608163208, −8.997188209433688262094213228938, −8.100557276558661501375151136729, −7.39500675848415527813322902401, −6.68337555070165749539635926378, −5.65673538214919461925099154589, −4.57772335230375983126653523610, −4.22308004814163926392777277835, −2.51676695330962227028664132581, −1.87141801462053587037809439727, 0.76618312973640331473668257967, 2.44073975174804887243873314855, 3.52001274082573167693286635440, 4.19604982427584580125755303700, 5.55181477251548754789703190542, 6.19341001085254545499462398235, 6.82127411470994223871723853581, 8.087037699184022222327052723812, 8.643143316780750637366533237565, 9.247111770411681965366853272028

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