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Properties

Label 2-40e2-80.29-c1-0-0
Degree $2$
Conductor $1600$
Sign $-0.992 + 0.121i$
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  + (0.0623 − 0.0623i)3-s − 0.375·7-s + 2.99i·9-s + (−2.36 + 2.36i)11-s + (−1.76 + 1.76i)13-s − 4.64i·17-s + (−2.34 − 2.34i)19-s + (−0.0234 + 0.0234i)21-s + 2.07·23-s + (0.373 + 0.373i)27-s + (−2.55 − 2.55i)29-s − 8.51·31-s + 0.295i·33-s + (−7.62 − 7.62i)37-s + 0.219i·39-s + ⋯
L(s)  = 1  + (0.0359 − 0.0359i)3-s − 0.142·7-s + 0.997i·9-s + (−0.713 + 0.713i)11-s + (−0.489 + 0.489i)13-s − 1.12i·17-s + (−0.539 − 0.539i)19-s + (−0.00511 + 0.00511i)21-s + 0.433·23-s + (0.0718 + 0.0718i)27-s + (−0.474 − 0.474i)29-s − 1.52·31-s + 0.0513i·33-s + (−1.25 − 1.25i)37-s + 0.0352i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.992 + 0.121i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09835170539\)
\(L(\frac12)\) \(\approx\) \(0.09835170539\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0623 + 0.0623i)T - 3iT^{2} \)
7 \( 1 + 0.375T + 7T^{2} \)
11 \( 1 + (2.36 - 2.36i)T - 11iT^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + 4.64iT - 17T^{2} \)
19 \( 1 + (2.34 + 2.34i)T + 19iT^{2} \)
23 \( 1 - 2.07T + 23T^{2} \)
29 \( 1 + (2.55 + 2.55i)T + 29iT^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 + (7.62 + 7.62i)T + 37iT^{2} \)
41 \( 1 + 3.77iT - 41T^{2} \)
43 \( 1 + (-6.21 - 6.21i)T + 43iT^{2} \)
47 \( 1 + 9.71iT - 47T^{2} \)
53 \( 1 + (3.03 + 3.03i)T + 53iT^{2} \)
59 \( 1 + (8.11 - 8.11i)T - 59iT^{2} \)
61 \( 1 + (-0.728 - 0.728i)T + 61iT^{2} \)
67 \( 1 + (0.969 - 0.969i)T - 67iT^{2} \)
71 \( 1 - 9.14iT - 71T^{2} \)
73 \( 1 - 7.56T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 3.86iT - 97T^{2} \)
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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

−9.678799826100547120108264876500, −9.174895104271200756265475729316, −8.146380860590326472510354545482, −7.29432716898739700927388448421, −6.93874577315449071000705979370, −5.47963413669092855286158231816, −4.99983322315819844880626067758, −4.03281866604945807166040993667, −2.66529234325091548974960394004, −1.96785576757625421090635895161, 0.03513718367214094994801254941, 1.62964553079020196268941106914, 3.07468820213272397212264122418, 3.67774314854666192822895427608, 4.88899608809559331473055439573, 5.82818035632890540964604935190, 6.43762431303088624647389112820, 7.47679325421339888554399178919, 8.237389036108221494643302834406, 8.987754649564956180950327655652

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