↑

Properties

Label 2-3400-680.403-c0-0-3
Degree $2$
Conductor $3400$
Sign $-0.998 + 0.0493i$
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  + (−0.923 − 0.382i)2-s + (−0.630 − 0.125i)3-s + (0.707 + 0.707i)4-s + (0.534 + 0.357i)6-s + (−0.382 − 0.923i)8-s + (−0.542 − 0.224i)9-s + (−0.996 + 1.49i)11-s + (−0.357 − 0.534i)12-s + i·16-s + (−0.608 − 0.793i)17-s + (0.414 + 0.414i)18-s + (1.78 − 0.739i)19-s + (1.49 − 0.996i)22-s + (0.125 + 0.630i)24-s + (0.848 + 0.566i)27-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (−0.630 − 0.125i)3-s + (0.707 + 0.707i)4-s + (0.534 + 0.357i)6-s + (−0.382 − 0.923i)8-s + (−0.542 − 0.224i)9-s + (−0.996 + 1.49i)11-s + (−0.357 − 0.534i)12-s + i·16-s + (−0.608 − 0.793i)17-s + (0.414 + 0.414i)18-s + (1.78 − 0.739i)19-s + (1.49 − 0.996i)22-s + (0.125 + 0.630i)24-s + (0.848 + 0.566i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-0.998 + 0.0493i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (2443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -0.998 + 0.0493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09755963745\)
\(L(\frac12)\) \(\approx\) \(0.09755963745\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 \)
17 \( 1 + (0.608 + 0.793i)T \)
good3 \( 1 + (0.630 + 0.125i)T + (0.923 + 0.382i)T^{2} \)
7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.996 - 1.49i)T + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (-1.78 + 0.739i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (1.95 - 0.389i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 + (0.860 + 0.860i)T + iT^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.491 + 0.735i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-1.83 - 0.758i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
97 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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

−8.438771064669981307655621988311, −7.69185582104157406655702794105, −6.99371590612665818777070485257, −6.52964037743258972111674511437, −5.21737658521445609519993744517, −4.85352831610882918286587013366, −3.34511462072512242830054530336, −2.65559842308830557497319412332, −1.57093907702973905689385139253, −0.087219834862242580278909498387, 1.33744415208380617872864279922, 2.67550106377499794285629440518, 3.48835714792255797249919686255, 5.09169400930439595170331752414, 5.50703210138140601912943775557, 6.11769744048917570854931891347, 6.92926132617551247011220014291, 7.914209539739145653614540952900, 8.339940827573582580331670724410, 8.946594031331771097302885547857

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