↑

Properties

Label 2-3400-680.643-c0-0-0
Degree $2$
Conductor $3400$
Sign $0.139 - 0.990i$
Analytic cond. $1.69682$
Root an. cond. $1.30262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−1.49 + 0.996i)3-s + (−0.707 − 0.707i)4-s + (−0.349 − 1.75i)6-s + (0.923 − 0.382i)8-s + (0.848 − 2.04i)9-s + (−1.85 + 0.369i)11-s + (1.75 + 0.349i)12-s + i·16-s + (−0.793 + 0.608i)17-s + (1.56 + 1.56i)18-s + (−0.739 − 1.78i)19-s + (0.369 − 1.85i)22-s + (−0.996 + 1.49i)24-s + (0.426 + 2.14i)27-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−1.49 + 0.996i)3-s + (−0.707 − 0.707i)4-s + (−0.349 − 1.75i)6-s + (0.923 − 0.382i)8-s + (0.848 − 2.04i)9-s + (−1.85 + 0.369i)11-s + (1.75 + 0.349i)12-s + i·16-s + (−0.793 + 0.608i)17-s + (1.56 + 1.56i)18-s + (−0.739 − 1.78i)19-s + (0.369 − 1.85i)22-s + (−0.996 + 1.49i)24-s + (0.426 + 2.14i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ 0.139 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3457610328\)
\(L(\frac12)\) \(\approx\) \(0.3457610328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 \)
17 \( 1 + (0.793 - 0.608i)T \)
good3 \( 1 + (1.49 - 0.996i)T + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (1.85 - 0.369i)T + (0.923 - 0.382i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (0.739 + 1.78i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (-1.09 - 0.732i)T + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.382 + 0.923i)T^{2} \)
67 \( 1 + (-1.12 - 1.12i)T + iT^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.630 - 0.125i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.184 + 0.184i)T + iT^{2} \)
97 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
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

−8.934120573384056149130073424422, −8.289285095995170098490988831861, −7.18059936731498770375770429623, −6.71995404727288650671979825170, −5.79506720587231825968108865500, −5.29116271081816525899262655908, −4.64362366507436112556208617373, −4.08020917593788002752233104333, −2.43603099425304147997719650633, −0.51875836478520350967402433247, 0.62030011632183629841378090373, 1.90938501523272955790493308297, 2.59308272008480590013283952982, 3.95816828688088966498768736369, 4.98544059907328660172761561289, 5.49836035391329830054053510848, 6.32416050042496800671352503277, 7.27501895670173475968129675165, 7.905694376678152490637540705925, 8.391078428362667082845082283345

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