↑

Properties

Label 2-104-8.5-c1-0-7
Degree $2$
Conductor $104$
Sign $-0.302 + 0.953i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (−1.40 − 0.144i)2-s − 2.10i·3-s + (1.95 + 0.406i)4-s − i·5-s + (−0.304 + 2.95i)6-s − 2.10·7-s + (−2.69 − 0.855i)8-s − 1.42·9-s + (−0.144 + 1.40i)10-s − 5.62i·11-s + (0.855 − 4.11i)12-s + i·13-s + (2.95 + 0.304i)14-s − 2.10·15-s + (3.66 + 1.59i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.994 − 0.102i)2-s − 1.21i·3-s + (0.979 + 0.203i)4-s − 0.447i·5-s + (−0.124 + 1.20i)6-s − 0.794·7-s + (−0.953 − 0.302i)8-s − 0.473·9-s + (−0.0457 + 0.444i)10-s − 1.69i·11-s + (0.246 − 1.18i)12-s + 0.277i·13-s + (0.790 + 0.0812i)14-s − 0.542·15-s + (0.917 + 0.398i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.302 + 0.953i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ -0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.372229 - 0.508623i\)
\(L(\frac12)\) \(\approx\) \(0.372229 - 0.508623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.144i)T \)
13 \( 1 - iT \)
good3 \( 1 + 2.10iT - 3T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 + 2.10T + 7T^{2} \)
11 \( 1 + 5.62iT - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 - 7.83iT - 29T^{2} \)
31 \( 1 - 9.62T + 31T^{2} \)
37 \( 1 + 0.421iT - 37T^{2} \)
41 \( 1 - 5.83T + 41T^{2} \)
43 \( 1 + 0.475iT - 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 + 4.57iT - 53T^{2} \)
59 \( 1 - 8.67iT - 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 + 9.15T + 71T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 - 8.20iT - 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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

−13.17005772535663301630164282477, −12.42386264715700305843013547958, −11.43951535242893302687591850785, −10.20786455988671614888931299836, −8.878676176043689530277242997170, −8.095576989957565454396671794270, −6.84323071734673711232921772970, −5.97718551646058476254798047040, −3.08655711008613058948118050814, −1.07910270735428926520743501194, 2.85254419398033786893069896852, 4.64411032090639821115518811692, 6.42610539009518128855872041978, 7.50646618882037522947646129135, 9.106442262082077098019735404860, 9.911760805431894764347143159163, 10.39684228635846845693092553961, 11.64875295878993443392550289566, 12.89592816902959101860939142850, 14.69300829993160884560845264179

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