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Properties

Label 2-40e2-40.27-c1-0-26
Degree $2$
Conductor $1600$
Sign $0.685 + 0.727i$
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  + (1.41 − 1.41i)3-s + (2.44 − 2.44i)7-s − 1.00i·9-s + 3.46·11-s + (2.82 + 2.82i)13-s + (4.89 + 4.89i)17-s − 3.46i·19-s − 6.92i·21-s + (−2.44 − 2.44i)23-s + (2.82 + 2.82i)27-s − 6.92·29-s + 8i·31-s + (4.89 − 4.89i)33-s + (−5.65 + 5.65i)37-s + 8.00·39-s + ⋯
L(s)  = 1  + (0.816 − 0.816i)3-s + (0.925 − 0.925i)7-s − 0.333i·9-s + 1.04·11-s + (0.784 + 0.784i)13-s + (1.18 + 1.18i)17-s − 0.794i·19-s − 1.51i·21-s + (−0.510 − 0.510i)23-s + (0.544 + 0.544i)27-s − 1.28·29-s + 1.43i·31-s + (0.852 − 0.852i)33-s + (−0.929 + 0.929i)37-s + 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.685 + 0.727i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.835362905\)
\(L(\frac12)\) \(\approx\) \(2.835362905\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \)
7 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
17 \( 1 + (-4.89 - 4.89i)T + 17iT^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (2.44 + 2.44i)T + 23iT^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 + (5.65 - 5.65i)T - 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (1.41 - 1.41i)T - 43iT^{2} \)
47 \( 1 + (-2.44 + 2.44i)T - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + (-1.41 - 1.41i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (4.89 - 4.89i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
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

−9.033577495893369660879472564989, −8.373343227731065401162232825355, −7.83298676268241645230549874519, −6.96585357476113589380891354693, −6.39872201004817072281279742085, −5.09234504536768066939045241732, −4.05817449069348479736244216318, −3.35322349959663623705139721757, −1.76185335130438538793288632464, −1.35385093537293206400236773557, 1.36980404452878574635545615403, 2.64181622890065607071503270419, 3.61143982064708789620121337137, 4.25808973490635060553578738023, 5.54394972964035351631034274945, 5.89540728983656201618113290988, 7.46891608780069290103219067877, 8.008424974270466136650230981637, 8.962966250944446924828975363057, 9.274638836191907159165646065152

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