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Properties

Label 2-40e2-16.13-c1-0-9
Degree $2$
Conductor $1600$
Sign $-0.871 - 0.491i$
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  + (2.16 + 2.16i)3-s + 3.30i·7-s + 6.40i·9-s + (−2.01 + 2.01i)11-s + (−0.794 − 0.794i)13-s − 4.61·17-s + (3.48 + 3.48i)19-s + (−7.16 + 7.16i)21-s − 7.99i·23-s + (−7.38 + 7.38i)27-s + (−1.95 − 1.95i)29-s + 5.12·31-s − 8.72·33-s + (−0.448 + 0.448i)37-s − 3.44i·39-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)3-s + 1.24i·7-s + 2.13i·9-s + (−0.606 + 0.606i)11-s + (−0.220 − 0.220i)13-s − 1.11·17-s + (0.800 + 0.800i)19-s + (−1.56 + 1.56i)21-s − 1.66i·23-s + (−1.42 + 1.42i)27-s + (−0.362 − 0.362i)29-s + 0.920·31-s − 1.51·33-s + (−0.0736 + 0.0736i)37-s − 0.551i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.871 - 0.491i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.871 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.218334415\)
\(L(\frac12)\) \(\approx\) \(2.218334415\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \)
7 \( 1 - 3.30iT - 7T^{2} \)
11 \( 1 + (2.01 - 2.01i)T - 11iT^{2} \)
13 \( 1 + (0.794 + 0.794i)T + 13iT^{2} \)
17 \( 1 + 4.61T + 17T^{2} \)
19 \( 1 + (-3.48 - 3.48i)T + 19iT^{2} \)
23 \( 1 + 7.99iT - 23T^{2} \)
29 \( 1 + (1.95 + 1.95i)T + 29iT^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + (0.448 - 0.448i)T - 37iT^{2} \)
41 \( 1 - 4.02iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 + (-3.35 + 3.35i)T - 53iT^{2} \)
59 \( 1 + (2.07 - 2.07i)T - 59iT^{2} \)
61 \( 1 + (0.557 + 0.557i)T + 61iT^{2} \)
67 \( 1 + (0.636 + 0.636i)T + 67iT^{2} \)
71 \( 1 - 6.85iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + (-9.48 - 9.48i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 + 0.709T + 97T^{2} \)
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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

−9.676676790974177438267755096600, −8.951148574529213937701998329251, −8.369479775608212169315851680586, −7.75111689585162867433829831295, −6.48165135203616491118051079782, −5.32499777644448801974704083271, −4.70460026542168440461984354137, −3.80613589107439842156874876380, −2.62752705384433101085269820811, −2.30435527408210947175720442927, 0.70284447896167170493704247035, 1.82474862432859966837998040692, 2.92615450296298730395036261117, 3.64137580344623156360022847971, 4.79886698309995995439172533291, 6.13123966023296919150606144527, 7.03055547452565928652589714556, 7.47860165925440692857344041161, 8.068762394294943951298416781529, 9.020003726619289540988499499481

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