↑

Properties

Label 1-680-680.379-r0-0-0
Degree $1$
Conductor $680$
Sign $0.163 + 0.986i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
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  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s − i·13-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.382 − 0.923i)23-s + (−0.923 + 0.382i)27-s + (−0.923 − 0.382i)29-s + (−0.382 + 0.923i)31-s + 33-s + (−0.382 + 0.923i)37-s + (0.923 + 0.382i)39-s + (−0.923 + 0.382i)41-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s − i·13-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.382 − 0.923i)23-s + (−0.923 + 0.382i)27-s + (−0.923 − 0.382i)29-s + (−0.382 + 0.923i)31-s + 33-s + (−0.382 + 0.923i)37-s + (0.923 + 0.382i)39-s + (−0.923 + 0.382i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6043845211 + 0.5123074992i\)
\(L(\frac12)\) \(\approx\) \(0.6043845211 + 0.5123074992i\)
\(L(1)\) \(\approx\) \(0.8958834694 + 0.02113658490i\)
\(L(1)\) \(\approx\) \(0.8958834694 + 0.02113658490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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

−22.26954337756617428156184393853, −21.92160189626595756614043400777, −20.90078173259856288114632641305, −20.03197834783220879881394316313, −19.53684854115303335640191621641, −18.66954476862502246323613491258, −17.301598905120376681707349567017, −16.77516090374792304814862582811, −15.81751597724565491422646768391, −15.32673601119151297016736137856, −14.2694403867876743658194176715, −13.47585689838080036047824454830, −12.74087529219762972488583771727, −11.36171691318209956646298862742, −10.71035544939059155271897877552, −9.83045674571804518215868464016, −9.096304515898984501425490475116, −8.24162657445486626538009149511, −7.16445096863116384733162856906, −5.96978410281245173577182269462, −5.225800722689767050281597620022, −3.78443061251514319444837554363, −3.47579555546119517841235096427, −2.25920159701676616335104148616, −0.34718932696993099703184997734, 1.534948958112068370774005653144, 2.345640865677365334857624953657, 3.452805589038426210392853661563, 4.513600711416783032155313109309, 6.02935050475304055984975593903, 6.59071296885801952079307863993, 7.40294137203033871839579885781, 8.51553568445069241629404794809, 9.26030851430790146431228953955, 10.08482032472979627251061271666, 11.42488201097791077204254021690, 12.396584619197248532123269289793, 12.67316992090670644937763246425, 13.79732202867003537105295596874, 14.53830433298055362983785465000, 15.33492293972743006380284072324, 16.490393232391420793284279925980, 17.16192653023985435240206904019, 18.2711589412610225291735520902, 18.83791633844966145534514179873, 19.54008126390012088459108242610, 20.30072247480739341893317830268, 21.17219281238601087263942780112, 22.32721627252999556530167317566, 22.91370808272665937424600817277

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