↑

Properties

Label 2-40e2-40.3-c1-0-0
Degree $2$
Conductor $1600$
Sign $-0.287 - 0.957i$
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  + (−0.456 − 0.456i)3-s + (−2.79 − 2.79i)7-s − 2.58i·9-s − 4.37·11-s + (1.73 − 1.73i)13-s + (−3 + 3i)17-s + 3.46i·19-s + 2.55i·21-s + (−0.791 + 0.791i)23-s + (−2.55 + 2.55i)27-s + 5.29·29-s + 1.58i·31-s + (1.99 + 1.99i)33-s + (5.19 + 5.19i)37-s − 1.58·39-s + ⋯
L(s)  = 1  + (−0.263 − 0.263i)3-s + (−1.05 − 1.05i)7-s − 0.860i·9-s − 1.31·11-s + (0.480 − 0.480i)13-s + (−0.727 + 0.727i)17-s + 0.794i·19-s + 0.556i·21-s + (−0.164 + 0.164i)23-s + (−0.490 + 0.490i)27-s + 0.982·29-s + 0.284i·31-s + (0.348 + 0.348i)33-s + (0.854 + 0.854i)37-s − 0.253·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2123149792\)
\(L(\frac12)\) \(\approx\) \(0.2123149792\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.456 + 0.456i)T + 3iT^{2} \)
7 \( 1 + (2.79 + 2.79i)T + 7iT^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
13 \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (0.791 - 0.791i)T - 23iT^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 1.58iT - 31T^{2} \)
37 \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \)
41 \( 1 + 7.58T + 41T^{2} \)
43 \( 1 + (-8.29 - 8.29i)T + 43iT^{2} \)
47 \( 1 + (-0.791 - 0.791i)T + 47iT^{2} \)
53 \( 1 + (-2.64 + 2.64i)T - 53iT^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 9.66iT - 61T^{2} \)
67 \( 1 + (8.29 - 8.29i)T - 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (0.582 + 0.582i)T + 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (4.83 + 4.83i)T + 83iT^{2} \)
89 \( 1 - 3.16iT - 89T^{2} \)
97 \( 1 + (0.582 - 0.582i)T - 97iT^{2} \)
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

−9.934364283415061723941212856111, −8.796562907196027710477564094229, −7.978900224767563478929066568222, −7.18669061524264068896582155180, −6.33691793937079642019763306948, −5.87103643422965104959665567649, −4.56073518681094418959194672885, −3.63540905327814781097094432246, −2.84303004610919970746286465905, −1.15880332474627391846915092695, 0.090986966732277607534472287152, 2.33607276951438416255648687226, 2.80407301239109361826728947618, 4.24206897643503425065348318589, 5.15142064331199565183697013256, 5.78941215565211766568641756963, 6.68641732677106022787743970030, 7.55451591964873349429062004438, 8.558250380695678907341522045812, 9.130325582903511075501320752165

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