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Properties

Label 2-40e2-80.43-c1-0-25
Degree $2$
Conductor $1600$
Sign $0.663 + 0.748i$
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  − 0.517i·3-s + (3.34 − 3.34i)7-s + 2.73·9-s + (1.09 − 1.09i)11-s + 4.89·13-s + (−0.707 + 0.707i)17-s + (−2.09 + 2.09i)19-s + (−1.73 − 1.73i)21-s + (4.38 + 4.38i)23-s − 2.96i·27-s + (−4.73 − 4.73i)29-s + 6.19i·31-s + (−0.568 − 0.568i)33-s − 6.03·37-s − 2.53i·39-s + ⋯
L(s)  = 1  − 0.298i·3-s + (1.26 − 1.26i)7-s + 0.910·9-s + (0.331 − 0.331i)11-s + 1.35·13-s + (−0.171 + 0.171i)17-s + (−0.481 + 0.481i)19-s + (−0.377 − 0.377i)21-s + (0.913 + 0.913i)23-s − 0.571i·27-s + (−0.878 − 0.878i)29-s + 1.11i·31-s + (−0.0989 − 0.0989i)33-s − 0.992·37-s − 0.406i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.316205526\)
\(L(\frac12)\) \(\approx\) \(2.316205526\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.517iT - 3T^{2} \)
7 \( 1 + (-3.34 + 3.34i)T - 7iT^{2} \)
11 \( 1 + (-1.09 + 1.09i)T - 11iT^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 + (0.707 - 0.707i)T - 17iT^{2} \)
19 \( 1 + (2.09 - 2.09i)T - 19iT^{2} \)
23 \( 1 + (-4.38 - 4.38i)T + 23iT^{2} \)
29 \( 1 + (4.73 + 4.73i)T + 29iT^{2} \)
31 \( 1 - 6.19iT - 31T^{2} \)
37 \( 1 + 6.03T + 37T^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 - 0.656T + 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 + (7.73 + 7.73i)T + 59iT^{2} \)
61 \( 1 + (3.19 - 3.19i)T - 61iT^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 + 0.928T + 71T^{2} \)
73 \( 1 + (-8.81 + 8.81i)T - 73iT^{2} \)
79 \( 1 - 2.19T + 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-11.5 + 11.5i)T - 97iT^{2} \)
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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

−9.212417045862336015518390893884, −8.360988908481661290152665756105, −7.68339192088184687709868093046, −7.02390169157776384162642077436, −6.18089703315652069602019591496, −5.07184746190633860347720581654, −4.14325254431431935504784561202, −3.57532341250881269850627576044, −1.69935127921607798180273928483, −1.13812648872713016424791088741, 1.40124506754617687162271518072, 2.29877182135251106157050524498, 3.67765584920306994526957284286, 4.62112414099570193542762938160, 5.25368962036860205722020989163, 6.24636346790676002059647074437, 7.10486488452770295453247776418, 8.095306012860804863045103515607, 8.864378522978917964226883053675, 9.212758210138621726270379115159

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