↑

Properties

Label 1-26e2-676.423-r0-0-0
Degree $1$
Conductor $676$
Sign $-0.304 - 0.952i$
Analytic cond. $3.13933$
Root an. cond. $3.13933$
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  + (−0.278 − 0.960i)3-s + (0.935 − 0.354i)5-s + (0.391 − 0.919i)7-s + (−0.845 + 0.534i)9-s + (0.534 − 0.845i)11-s + (−0.600 − 0.799i)15-s + (0.919 + 0.391i)17-s + (0.866 + 0.5i)19-s + (−0.992 − 0.120i)21-s + (−0.5 − 0.866i)23-s + (0.748 − 0.663i)25-s + (0.748 + 0.663i)27-s + (−0.0402 − 0.999i)29-s + (−0.663 + 0.748i)31-s + (−0.960 − 0.278i)33-s + ⋯
L(s)  = 1  + (−0.278 − 0.960i)3-s + (0.935 − 0.354i)5-s + (0.391 − 0.919i)7-s + (−0.845 + 0.534i)9-s + (0.534 − 0.845i)11-s + (−0.600 − 0.799i)15-s + (0.919 + 0.391i)17-s + (0.866 + 0.5i)19-s + (−0.992 − 0.120i)21-s + (−0.5 − 0.866i)23-s + (0.748 − 0.663i)25-s + (0.748 + 0.663i)27-s + (−0.0402 − 0.999i)29-s + (−0.663 + 0.748i)31-s + (−0.960 − 0.278i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ -0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9733913597 - 1.333666179i\)
\(L(\frac12)\) \(\approx\) \(0.9733913597 - 1.333666179i\)
\(L(1)\) \(\approx\) \(1.065654091 - 0.6226200668i\)
\(L(1)\) \(\approx\) \(1.065654091 - 0.6226200668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.278 - 0.960i)T \)
5 \( 1 + (0.935 - 0.354i)T \)
7 \( 1 + (0.391 - 0.919i)T \)
11 \( 1 + (0.534 - 0.845i)T \)
17 \( 1 + (0.919 + 0.391i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.0402 - 0.999i)T \)
31 \( 1 + (-0.663 + 0.748i)T \)
37 \( 1 + (0.979 + 0.200i)T \)
41 \( 1 + (-0.960 + 0.278i)T \)
43 \( 1 + (-0.200 - 0.979i)T \)
47 \( 1 + (0.822 + 0.568i)T \)
53 \( 1 + (0.120 + 0.992i)T \)
59 \( 1 + (-0.160 + 0.987i)T \)
61 \( 1 + (0.799 + 0.600i)T \)
67 \( 1 + (-0.0804 + 0.996i)T \)
71 \( 1 + (-0.721 - 0.692i)T \)
73 \( 1 + (-0.464 - 0.885i)T \)
79 \( 1 + (-0.568 + 0.822i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.774 - 0.632i)T \)
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

−22.71584122528089666153164821280, −21.89696002820746021439761669078, −21.69423295483848888429799029253, −20.630453271363647700646311277345, −20.05402614969961223162294956094, −18.65183305402595157470544544879, −17.94984267952964164595062086034, −17.32811891512258897274989028875, −16.40470999113274559118207956243, −15.51744086352151547565623724570, −14.65889607668616943936964678702, −14.24849639144258508380370956440, −12.97125669686016457657999791066, −11.856042236408381845096805756971, −11.329144810133281095279561339549, −10.12503727902521202524586780641, −9.54804524676430115970473421427, −8.952903780163316489102783525667, −7.59064010371538080182304582861, −6.42742969598432127951973419310, −5.45708563374947942306706134836, −5.02433324010057059775328852445, −3.65504380250800629495374792413, −2.665057957283971189466971082846, −1.547568136040977861579179829224, 0.93810300980662079305769782954, 1.56068035121887152242737269516, 2.854472773918336699334407008625, 4.1431682796840769602175706556, 5.45542033239680939705184753997, 6.0253365698186629052669854031, 7.00819206393716720093368296667, 7.948301806685914409535358571814, 8.723904729180857327399788384270, 9.94792353622300515475751537509, 10.71760980335635479740840073775, 11.749695819718402095693127848653, 12.49260230727312138954581192001, 13.56403672439146725487805353231, 13.912871042985715639777365802300, 14.66287035660983692844682667919, 16.56743636964155792524963019298, 16.64673404443715901263005404066, 17.59913864594969047940011366733, 18.32799198627066989600515401160, 19.121630443125273757078072837747, 20.13251707663656982876664738659, 20.725438217152384962357228979996, 21.78875153665507812180021327374, 22.50721743510271891737680355987

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