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Properties

Label 2-40e2-20.7-c1-0-5
Degree $2$
Conductor $1600$
Sign $-0.608 - 0.793i$
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  + (1.58 + 1.58i)3-s + 2.00i·9-s + 3.87i·11-s + (−2.44 + 2.44i)13-s + (−1.22 − 1.22i)17-s − 3.87·19-s + (3.16 + 3.16i)23-s + (1.58 − 1.58i)27-s + 6i·29-s + 7.74i·31-s + (−6.12 + 6.12i)33-s + (−4.89 − 4.89i)37-s − 7.74·39-s − 3·41-s + (3.16 − 3.16i)47-s + ⋯
L(s)  = 1  + (0.912 + 0.912i)3-s + 0.666i·9-s + 1.16i·11-s + (−0.679 + 0.679i)13-s + (−0.297 − 0.297i)17-s − 0.888·19-s + (0.659 + 0.659i)23-s + (0.304 − 0.304i)27-s + 1.11i·29-s + 1.39i·31-s + (−1.06 + 1.06i)33-s + (−0.805 − 0.805i)37-s − 1.24·39-s − 0.468·41-s + (0.461 − 0.461i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862206359\)
\(L(\frac12)\) \(\approx\) \(1.862206359\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 1.58i)T + 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + 17iT^{2} \)
19 \( 1 + 3.87T + 19T^{2} \)
23 \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 7.74iT - 31T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \)
53 \( 1 + (2.44 - 2.44i)T - 53iT^{2} \)
59 \( 1 - 7.74T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (4.74 - 4.74i)T - 67iT^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (1.58 + 1.58i)T + 83iT^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \)
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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

−9.495213284077708532910612899346, −9.075697971311250381217223472904, −8.358292298542017675360324740876, −7.20173022074283986826266053529, −6.78144281840543010593829532959, −5.25331765473117480835970958690, −4.59372111536168416162242789308, −3.81057728120043960247904399083, −2.80985903649396970181983383971, −1.82137606757472024392519464420, 0.60932393319578629990833750474, 2.08956810449491771749331902660, 2.81049377305404378584507015731, 3.81115659019223745677181382222, 5.01536696286629692137244317052, 6.07275162320236583241615891306, 6.79786978012972287295188210706, 7.70605562903103976076338705674, 8.340677050116216064772080137647, 8.768357297600401486423628841141

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