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Properties

Label 2-40e2-80.69-c1-0-16
Degree $2$
Conductor $1600$
Sign $0.999 + 0.00745i$
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  + (−0.488 − 0.488i)3-s + 4.71·7-s − 2.52i·9-s + (3.91 + 3.91i)11-s + (0.0878 + 0.0878i)13-s + 4.67i·17-s + (1.81 − 1.81i)19-s + (−2.30 − 2.30i)21-s + 1.63·23-s + (−2.69 + 2.69i)27-s + (−3.26 + 3.26i)29-s + 2.12·31-s − 3.82i·33-s + (−3.97 + 3.97i)37-s − 0.0858i·39-s + ⋯
L(s)  = 1  + (−0.282 − 0.282i)3-s + 1.78·7-s − 0.840i·9-s + (1.17 + 1.17i)11-s + (0.0243 + 0.0243i)13-s + 1.13i·17-s + (0.415 − 0.415i)19-s + (−0.502 − 0.502i)21-s + 0.339·23-s + (−0.519 + 0.519i)27-s + (−0.606 + 0.606i)29-s + 0.382·31-s − 0.665i·33-s + (−0.653 + 0.653i)37-s − 0.0137i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.999 + 0.00745i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.999 + 0.00745i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084169902\)
\(L(\frac12)\) \(\approx\) \(2.084169902\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.488 + 0.488i)T + 3iT^{2} \)
7 \( 1 - 4.71T + 7T^{2} \)
11 \( 1 + (-3.91 - 3.91i)T + 11iT^{2} \)
13 \( 1 + (-0.0878 - 0.0878i)T + 13iT^{2} \)
17 \( 1 - 4.67iT - 17T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 + (3.26 - 3.26i)T - 29iT^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 + (3.97 - 3.97i)T - 37iT^{2} \)
41 \( 1 + 8.25iT - 41T^{2} \)
43 \( 1 + (2.27 - 2.27i)T - 43iT^{2} \)
47 \( 1 - 4.06iT - 47T^{2} \)
53 \( 1 + (5.03 - 5.03i)T - 53iT^{2} \)
59 \( 1 + (5.16 + 5.16i)T + 59iT^{2} \)
61 \( 1 + (-7.12 + 7.12i)T - 61iT^{2} \)
67 \( 1 + (-7.49 - 7.49i)T + 67iT^{2} \)
71 \( 1 + 4.54iT - 71T^{2} \)
73 \( 1 + 8.30T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (1.16 + 1.16i)T + 83iT^{2} \)
89 \( 1 + 3.24iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.252499259094909955651289114477, −8.675868195042873878531708218794, −7.73118161396279601390438998623, −7.02549071899323986076150541842, −6.26519276078795640323853859398, −5.19764969008805369693960910111, −4.46322341084658327538879884147, −3.59662479283307787782125896931, −1.89346727625711580446542061603, −1.26686859760065309716999775819, 1.06055400842787863970999252526, 2.16204972890977282422492531453, 3.54739586151840004201069956507, 4.57766805774227470137475965419, 5.19370106162561298058013174558, 5.95785968158158403022960664291, 7.13756265574311906915110690524, 7.931338770147489206947133782688, 8.522062584781126329895882133930, 9.332055285794210801105505390615

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