↑

Properties

Label 2-40e2-80.27-c1-0-18
Degree $2$
Conductor $1600$
Sign $0.161 + 0.986i$
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  − 0.790·3-s + (−0.139 − 0.139i)7-s − 2.37·9-s + (−2.94 + 2.94i)11-s + 0.235i·13-s + (2.06 + 2.06i)17-s + (2.55 − 2.55i)19-s + (0.110 + 0.110i)21-s + (4.62 − 4.62i)23-s + 4.24·27-s + (−6.66 − 6.66i)29-s + 3.43i·31-s + (2.32 − 2.32i)33-s + 1.38i·37-s − 0.186i·39-s + ⋯
L(s)  = 1  − 0.456·3-s + (−0.0528 − 0.0528i)7-s − 0.791·9-s + (−0.888 + 0.888i)11-s + 0.0653i·13-s + (0.499 + 0.499i)17-s + (0.585 − 0.585i)19-s + (0.0241 + 0.0241i)21-s + (0.964 − 0.964i)23-s + 0.817·27-s + (−1.23 − 1.23i)29-s + 0.616i·31-s + (0.405 − 0.405i)33-s + 0.227i·37-s − 0.0298i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.161 + 0.986i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8556316361\)
\(L(\frac12)\) \(\approx\) \(0.8556316361\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.790T + 3T^{2} \)
7 \( 1 + (0.139 + 0.139i)T + 7iT^{2} \)
11 \( 1 + (2.94 - 2.94i)T - 11iT^{2} \)
13 \( 1 - 0.235iT - 13T^{2} \)
17 \( 1 + (-2.06 - 2.06i)T + 17iT^{2} \)
19 \( 1 + (-2.55 + 2.55i)T - 19iT^{2} \)
23 \( 1 + (-4.62 + 4.62i)T - 23iT^{2} \)
29 \( 1 + (6.66 + 6.66i)T + 29iT^{2} \)
31 \( 1 - 3.43iT - 31T^{2} \)
37 \( 1 - 1.38iT - 37T^{2} \)
41 \( 1 + 8.26iT - 41T^{2} \)
43 \( 1 + 5.40iT - 43T^{2} \)
47 \( 1 + (-6.84 + 6.84i)T - 47iT^{2} \)
53 \( 1 - 8.19T + 53T^{2} \)
59 \( 1 + (4.32 + 4.32i)T + 59iT^{2} \)
61 \( 1 + (9.15 - 9.15i)T - 61iT^{2} \)
67 \( 1 - 5.00iT - 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 + (11.3 + 11.3i)T + 73iT^{2} \)
79 \( 1 + 4.44T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 5.84T + 89T^{2} \)
97 \( 1 + (0.515 + 0.515i)T + 97iT^{2} \)
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

−9.149007003969221252241045410880, −8.519823425289149771705592666309, −7.49352849067842439711042671437, −6.92788681610822456603990950370, −5.76288635407557639565144602166, −5.27266826665693543779392165007, −4.29748544401326470387953858294, −3.09061800889529830533516707244, −2.11600325255981858813140179490, −0.39804392422348960783568050754, 1.11214665286011401181459316013, 2.80291224170447657761814527699, 3.40330182387980793080375645521, 4.85838856537607661159530047719, 5.61170902708986857000761617037, 6.04148641219394360378752904081, 7.36004611852973508338746153703, 7.86577582246004221036521764287, 8.875425009502752646119594027419, 9.499504970121601381269384890473

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