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Properties

Label 2-40e2-80.69-c1-0-7
Degree $2$
Conductor $1600$
Sign $0.939 - 0.341i$
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  + (−1.66 − 1.66i)3-s + 2.89·7-s + 2.53i·9-s + (−1.84 − 1.84i)11-s + (3.08 + 3.08i)13-s + 7.29i·17-s + (−1.23 + 1.23i)19-s + (−4.81 − 4.81i)21-s + 4.60·23-s + (−0.772 + 0.772i)27-s + (−4.24 + 4.24i)29-s − 2.06·31-s + 6.13i·33-s + (1.17 − 1.17i)37-s − 10.2i·39-s + ⋯
L(s)  = 1  + (−0.960 − 0.960i)3-s + 1.09·7-s + 0.845i·9-s + (−0.556 − 0.556i)11-s + (0.854 + 0.854i)13-s + 1.77i·17-s + (−0.283 + 0.283i)19-s + (−1.05 − 1.05i)21-s + 0.960·23-s + (−0.148 + 0.148i)27-s + (−0.788 + 0.788i)29-s − 0.370·31-s + 1.06i·33-s + (0.193 − 0.193i)37-s − 1.64i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170894462\)
\(L(\frac12)\) \(\approx\) \(1.170894462\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 1.66i)T + 3iT^{2} \)
7 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 + (1.84 + 1.84i)T + 11iT^{2} \)
13 \( 1 + (-3.08 - 3.08i)T + 13iT^{2} \)
17 \( 1 - 7.29iT - 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 - 4.60T + 23T^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
37 \( 1 + (-1.17 + 1.17i)T - 37iT^{2} \)
41 \( 1 - 4.61iT - 41T^{2} \)
43 \( 1 + (3.03 - 3.03i)T - 43iT^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + (-2.73 + 2.73i)T - 53iT^{2} \)
59 \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \)
67 \( 1 + (8.24 + 8.24i)T + 67iT^{2} \)
71 \( 1 - 3.25iT - 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + 0.113T + 79T^{2} \)
83 \( 1 + (-9.76 - 9.76i)T + 83iT^{2} \)
89 \( 1 - 3.74iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.323029251032012531033228041471, −8.392008350812906501900791134455, −7.907284757903145445060251703554, −6.92324407045811399832277457810, −6.17766955281038704349751782557, −5.58064356221645200787070484662, −4.62322617610294939146984578844, −3.53279054667599399093801479961, −1.87724769195767739705848516259, −1.21644276019525927731715650374, 0.58336863909619320283499184616, 2.24894717777425188379972189351, 3.57517328978854160938414938962, 4.67095241945808013209043010757, 5.14157990448755785237478239328, 5.73117046956489110911888155773, 6.99149763016870359341528347135, 7.73676118862353770636159304872, 8.663457524807647294480985587636, 9.505869741889067641901045847663

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