↑

Properties

Label 2-40e2-80.29-c1-0-32
Degree $2$
Conductor $1600$
Sign $-0.265 + 0.963i$
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 + 0.982·7-s − 7.82i·9-s + (1.62 − 1.62i)11-s + (−0.690 + 0.690i)13-s − 2.19i·17-s + (1.92 + 1.92i)19-s + (2.28 − 2.28i)21-s + 2.01·23-s + (−11.2 − 11.2i)27-s + (5.27 + 5.27i)29-s − 0.435·31-s − 7.56i·33-s + (−5.79 − 5.79i)37-s + 3.21i·39-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)3-s + 0.371·7-s − 2.60i·9-s + (0.490 − 0.490i)11-s + (−0.191 + 0.191i)13-s − 0.532i·17-s + (0.441 + 0.441i)19-s + (0.498 − 0.498i)21-s + 0.420·23-s + (−2.15 − 2.15i)27-s + (0.978 + 0.978i)29-s − 0.0781·31-s − 1.31i·33-s + (−0.953 − 0.953i)37-s + 0.514i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.265 + 0.963i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.265 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.869313608\)
\(L(\frac12)\) \(\approx\) \(2.869313608\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.982T + 7T^{2} \)
11 \( 1 + (-1.62 + 1.62i)T - 11iT^{2} \)
13 \( 1 + (0.690 - 0.690i)T - 13iT^{2} \)
17 \( 1 + 2.19iT - 17T^{2} \)
19 \( 1 + (-1.92 - 1.92i)T + 19iT^{2} \)
23 \( 1 - 2.01T + 23T^{2} \)
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 \( 1 - 9.21iT - 47T^{2} \)
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 \( 1 - 9.24T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-0.683 + 0.683i)T - 83iT^{2} \)
89 \( 1 - 5.44iT - 89T^{2} \)
97 \( 1 - 5.54iT - 97T^{2} \)
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

−9.044282372883707580905198691479, −8.290957975147112583415940650836, −7.66617941329843711462686489602, −6.92501158838746768712195836961, −6.28752830658204300166654447398, −5.05416396828024251787395074432, −3.70251095024818913112587569399, −2.99194569712086719836085525510, −1.94190203993209339034739268882, −1.00137010374805798156404426140, 1.80121243928520042862007409856, 2.87834478007263787614097652054, 3.65623325737599543731084275649, 4.61405029229664778605123883520, 5.05910381227210619564739348391, 6.46593377914901658577065455978, 7.60350064869688002097239038030, 8.226983187932245679675636206316, 8.910112772347956730871243677006, 9.628281586685689004240537745415

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