↑

Properties

Label 2-104-8.5-c1-0-2
Degree $2$
Conductor $104$
Sign $-0.844 - 0.535i$
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  + (−0.264 + 1.38i)2-s + 3.24i·3-s + (−1.85 − 0.735i)4-s − i·5-s + (−4.51 − 0.859i)6-s + 3.24·7-s + (1.51 − 2.38i)8-s − 7.55·9-s + (1.38 + 0.264i)10-s − 1.05i·11-s + (2.38 − 6.04i)12-s + i·13-s + (−0.859 + 4.51i)14-s + 3.24·15-s + (2.91 + 2.73i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + 1.87i·3-s + (−0.929 − 0.367i)4-s − 0.447i·5-s + (−1.84 − 0.351i)6-s + 1.22·7-s + (0.535 − 0.844i)8-s − 2.51·9-s + (0.439 + 0.0836i)10-s − 0.319i·11-s + (0.689 − 1.74i)12-s + 0.277i·13-s + (−0.229 + 1.20i)14-s + 0.838·15-s + (0.729 + 0.683i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.844 - 0.535i$
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: \((2,\ 104,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252785 + 0.871282i\)
\(L(\frac12)\) \(\approx\) \(0.252785 + 0.871282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 - 1.38i)T \)
13 \( 1 - iT \)
good3 \( 1 - 3.24iT - 3T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 + 1.05iT - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 2.94T + 23T^{2} \)
29 \( 1 + 7.43iT - 29T^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 + 6.55iT - 37T^{2} \)
41 \( 1 + 9.43T + 41T^{2} \)
43 \( 1 - 0.307iT - 43T^{2} \)
47 \( 1 - 6.80T + 47T^{2} \)
53 \( 1 - 1.55iT - 53T^{2} \)
59 \( 1 - 5.67iT - 59T^{2} \)
61 \( 1 + 9.67iT - 61T^{2} \)
67 \( 1 - 1.50iT - 67T^{2} \)
71 \( 1 + 5.36T + 71T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + 6.73T + 79T^{2} \)
83 \( 1 + 2.49iT - 83T^{2} \)
89 \( 1 - 2.11T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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

−14.51965518716233837383213018670, −13.84853064394682477955743064424, −11.88851441429741975113757829580, −10.68931236799145382940740107589, −9.804727822595036461101885921718, −8.739007260464816348533238368050, −8.031932151833472607456771422226, −5.85407368672292421061399252591, −4.89218619783730586542640453927, −4.01432896144831613757229661352, 1.42380722173675080461927282563, 2.76799505307778863101070137842, 5.11985936614830746946846781561, 6.87031571547776783271903673310, 7.924981903312673000645191797516, 8.717159788648226078400297696914, 10.56451157821253107144574533264, 11.54939746476429840077928129936, 12.18330158658257515878540548209, 13.20960711878510305467928698949

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