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Properties

Label 2-40e2-5.2-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.525 - 0.850i$
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 + (−1 + i)13-s + (1 + i)17-s + (1 + i)37-s − i·49-s + (1 − i)53-s + (−1 + i)73-s − 81-s + (−1 − i)97-s + 2·101-s + 2i·109-s + (1 − i)113-s + (−1 − i)117-s + ⋯
L(s)  = 1  + i·9-s + (−1 + i)13-s + (1 + i)17-s + (1 + i)37-s − i·49-s + (1 − i)53-s + (−1 + i)73-s − 81-s + (−1 − i)97-s + 2·101-s + 2i·109-s + (1 − i)113-s + (−1 − i)117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039512545\)
\(L(\frac12)\) \(\approx\) \(1.039512545\)
\(L(1)\) 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 \( 1 + T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.961734970081460408075153894472, −8.876564417090761194257583351309, −8.091247731466074846928739283111, −7.41883908849089870645206731706, −6.56712806286335577060941872781, −5.56364618365182883556818772206, −4.78836929313519103417948617163, −3.90170104291048891170229486744, −2.64759275157110111882515957514, −1.66118871882881372317107720211, 0.860198330781881761685279441821, 2.58330657180352164610614307723, 3.36643280550921688563021665879, 4.48220157498074805728549511348, 5.44700313966039662178670357305, 6.13321397127507000793770628340, 7.29226262087773670281017124505, 7.66579803539722451925421093393, 8.798436193317607234406669320793, 9.548399693725478716586461596171

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