Properties

Label 2-40e2-16.5-c1-0-3
Degree $2$
Conductor $1600$
Sign $-0.981 - 0.193i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 + 2.32i)3-s − 0.982i·7-s − 7.82i·9-s + (1.62 + 1.62i)11-s + (0.690 − 0.690i)13-s + 2.19·17-s + (−1.92 + 1.92i)19-s + (2.28 + 2.28i)21-s + 2.01i·23-s + (11.2 + 11.2i)27-s + (−5.27 + 5.27i)29-s − 0.435·31-s − 7.56·33-s + (5.79 + 5.79i)37-s + 3.21i·39-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)3-s − 0.371i·7-s − 2.60i·9-s + (0.490 + 0.490i)11-s + (0.191 − 0.191i)13-s + 0.532·17-s + (−0.441 + 0.441i)19-s + (0.498 + 0.498i)21-s + 0.420i·23-s + (2.15 + 2.15i)27-s + (−0.978 + 0.978i)29-s − 0.0781·31-s − 1.31·33-s + (0.953 + 0.953i)37-s + 0.514i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.981 - 0.193i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.981 - 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6088873993\)
\(L(\frac12)\) \(\approx\) \(0.6088873993\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 2.32i)T - 3iT^{2} \)
7 \( 1 + 0.982iT - 7T^{2} \)
11 \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \)
13 \( 1 + (-0.690 + 0.690i)T - 13iT^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
19 \( 1 + (1.92 - 1.92i)T - 19iT^{2} \)
23 \( 1 - 2.01iT - 23T^{2} \)
29 \( 1 + (5.27 - 5.27i)T - 29iT^{2} \)
31 \( 1 + 0.435T + 31T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (0.507 + 0.507i)T + 43iT^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 + (6.29 + 6.29i)T + 53iT^{2} \)
59 \( 1 + (-5.67 - 5.67i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (-4.53 + 4.53i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 - 9.24iT - 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (0.683 - 0.683i)T - 83iT^{2} \)
89 \( 1 - 5.44iT - 89T^{2} \)
97 \( 1 + 5.54T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.826111087400369196589681074933, −9.409853985625967061455820433365, −8.326308177981786550527373819374, −7.14619853456450462743453301164, −6.34331409605662729333939444422, −5.60014319136424966059451572927, −4.82588586250590335237667402813, −4.05345269336132128909711948091, −3.31403295326408280917757148810, −1.27806931316978217599909850511, 0.31703204652175626910725345184, 1.51096480133520742044665413198, 2.54485177737414093906176184831, 4.12238028992274666891122400218, 5.23991348218712708713149098238, 5.94079204733758018017978448896, 6.46658743091078986498419670518, 7.31135225930243460666721856782, 8.011025591350059599235460900757, 8.892369681807902297449018878650

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