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Properties

Label 2-40e2-80.29-c1-0-21
Degree $2$
Conductor $1600$
Sign $0.960 + 0.278i$
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  + (0.120 − 0.120i)3-s + 2.66·7-s + 2.97i·9-s + (3.49 − 3.49i)11-s + (2.94 − 2.94i)13-s + 1.85i·17-s + (−3.44 − 3.44i)19-s + (0.320 − 0.320i)21-s + 0.707·23-s + (0.716 + 0.716i)27-s + (3.49 + 3.49i)29-s − 6.84·31-s − 0.839i·33-s + (−0.0975 − 0.0975i)37-s − 0.705i·39-s + ⋯
L(s)  = 1  + (0.0692 − 0.0692i)3-s + 1.00·7-s + 0.990i·9-s + (1.05 − 1.05i)11-s + (0.815 − 0.815i)13-s + 0.448i·17-s + (−0.791 − 0.791i)19-s + (0.0698 − 0.0698i)21-s + 0.147·23-s + (0.137 + 0.137i)27-s + (0.649 + 0.649i)29-s − 1.22·31-s − 0.146i·33-s + (−0.0160 − 0.0160i)37-s − 0.113i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.960 + 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.156808638\)
\(L(\frac12)\) \(\approx\) \(2.156808638\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.120 + 0.120i)T - 3iT^{2} \)
7 \( 1 - 2.66T + 7T^{2} \)
11 \( 1 + (-3.49 + 3.49i)T - 11iT^{2} \)
13 \( 1 + (-2.94 + 2.94i)T - 13iT^{2} \)
17 \( 1 - 1.85iT - 17T^{2} \)
19 \( 1 + (3.44 + 3.44i)T + 19iT^{2} \)
23 \( 1 - 0.707T + 23T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 + 6.84T + 31T^{2} \)
37 \( 1 + (0.0975 + 0.0975i)T + 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-4.43 - 4.43i)T + 43iT^{2} \)
47 \( 1 - 1.89iT - 47T^{2} \)
53 \( 1 + (-7.43 - 7.43i)T + 53iT^{2} \)
59 \( 1 + (-0.959 + 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (-3.49 + 3.49i)T - 67iT^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 + (-3.87 + 3.87i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 4.79iT - 97T^{2} \)
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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

−9.022851115391356298760735887259, −8.605088791093729723864488438840, −7.936983433376852748297111961170, −7.03629203604235060487559439055, −6.02102191946363488380072656793, −5.31186687809044858294712044435, −4.34548265328431881233180979454, −3.41224809569852236582619312522, −2.15439067707130416136750578313, −1.04379013333349496456286442736, 1.22931273279867507415320561902, 2.15139107831571790047490467592, 3.79643319888785748080939820713, 4.20298269820760834601169036733, 5.26996663054864754635759908350, 6.43701643004124363288220632268, 6.84845078327934123615408836226, 7.958015554310012856742959616153, 8.718587676644379545937410478910, 9.386683465672859056190047157170

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