↑

Properties

Label 1-308-308.39-r0-0-0
Degree $1$
Conductor $308$
Sign $0.968 - 0.248i$
Analytic cond. $1.43034$
Root an. cond. $1.43034$
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  + (−0.669 + 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.913 − 0.406i)31-s + (0.669 + 0.743i)37-s + (−0.104 + 0.994i)39-s + (−0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.913 − 0.406i)31-s + (0.669 + 0.743i)37-s + (−0.104 + 0.994i)39-s + (−0.309 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(1.43034\)
Root analytic conductor: \(1.43034\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155408668 - 0.1456695447i\)
\(L(\frac12)\) \(\approx\) \(1.155408668 - 0.1456695447i\)
\(L(1)\) \(\approx\) \(1.019677403 + 0.01367592216i\)
\(L(1)\) \(\approx\) \(1.019677403 + 0.01367592216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.43189173295914826835252221963, −24.350931637302753519427757014875, −23.47561312217375686160475514620, −22.79741073465965633466104288987, −21.63203637122035525011506937693, −21.212413919862154242326189149911, −19.662200569166209345113427517565, −18.81505543868520505763971827710, −18.09975815855968987844641794537, −17.19787580971050058576227305819, −16.597780191084983037139634574655, −15.200293922886616230683828172477, −14.10349846867974347582278633778, −13.27505902978632258766466269416, −12.5531715677796900559106454912, −11.22901193317063201158162463243, −10.673457687007859045759017249970, −9.43755662703877352755666225199, −8.24513040202557190758578444612, −7.027570819948156804815267243181, −6.17502388558045122963006739491, −5.5011331305683675518721615273, −3.95926084249460311540916513040, −2.30926713279870153349578731927, −1.410304711266756693826541639826, 0.93281164761575183835757144365, 2.6186660369062577642237657524, 4.04009943068587848933811306747, 5.13975634048191018374096576453, 5.87520126409330384343154706356, 6.91469063144959557289194235326, 8.63339860881646546007585479444, 9.3343721869267000383996684995, 10.4223092268964429682931371801, 11.051343419282272000985866710, 12.35648812120132826109566868124, 13.16710675220482331434354249630, 14.316115484084504786438394207996, 15.31696276795108723091103581470, 16.334428845002729587914035171728, 16.96148551196433776628280222470, 17.89509245062734879190001448564, 18.626956057027626793478757868658, 20.35849781530034954092992486244, 20.72126468183941925253839326416, 21.72064235811277212210827868696, 22.43465874840967290582443579486, 23.34045877772706210828073866944, 24.29058549708295852425952815104, 25.4271980085014114591314702073

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