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Properties

Label 2-40e2-80.67-c1-0-12
Degree $2$
Conductor $1600$
Sign $-0.606 - 0.795i$
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  + 2.55i·3-s + (2.40 + 2.40i)7-s − 3.51·9-s + (2.67 + 2.67i)11-s + 2.40·13-s + (0.0750 + 0.0750i)17-s + (2.67 + 2.67i)19-s + (−6.13 + 6.13i)21-s + (2.12 − 2.12i)23-s − 1.30i·27-s + (3.95 − 3.95i)29-s − 1.65i·31-s + (−6.83 + 6.83i)33-s + 2.53·37-s + 6.12i·39-s + ⋯
L(s)  = 1  + 1.47i·3-s + (0.908 + 0.908i)7-s − 1.17·9-s + (0.807 + 0.807i)11-s + 0.666·13-s + (0.0182 + 0.0182i)17-s + (0.613 + 0.613i)19-s + (−1.33 + 1.33i)21-s + (0.442 − 0.442i)23-s − 0.250i·27-s + (0.734 − 0.734i)29-s − 0.297i·31-s + (−1.18 + 1.18i)33-s + 0.416·37-s + 0.981i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.095349525\)
\(L(\frac12)\) \(\approx\) \(2.095349525\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.55iT - 3T^{2} \)
7 \( 1 + (-2.40 - 2.40i)T + 7iT^{2} \)
11 \( 1 + (-2.67 - 2.67i)T + 11iT^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
17 \( 1 + (-0.0750 - 0.0750i)T + 17iT^{2} \)
19 \( 1 + (-2.67 - 2.67i)T + 19iT^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 + (-3.95 + 3.95i)T - 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 + 1.70iT - 41T^{2} \)
43 \( 1 + 3.84T + 43T^{2} \)
47 \( 1 + (2.15 - 2.15i)T - 47iT^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 + (-5.29 + 5.29i)T - 59iT^{2} \)
61 \( 1 + (-10.2 - 10.2i)T + 61iT^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (9.99 + 9.99i)T + 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (5.00 + 5.00i)T + 97iT^{2} \)
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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

−9.801577506102537371526017194993, −8.867767772192534539308679322413, −8.498704838506763798022064903407, −7.39496722968743518316690505893, −6.20412982127458995534200309667, −5.41029811660448747268201075562, −4.60709325394532339393998612281, −4.01081167042836904430888976174, −2.90240899740410201085414205064, −1.60155779394571877849238370578, 0.976099887221633645513815925698, 1.44273109143610074378698187718, 2.90040427677226204911953060441, 3.97537330512082234474586022656, 5.10364523443998634608977810062, 6.10535407876597940272319371097, 6.90683338877521418748652841538, 7.37149596457478128600845726718, 8.308002897866191201225869834066, 8.729781171112374114722302134521

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