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Properties

Label 2-40e2-40.37-c0-0-4
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: $\chi_{1600} (1057, \cdot )$
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

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

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