↑

Properties

Label 1-680-680.117-r1-0-0
Degree $1$
Conductor $680$
Sign $-0.953 + 0.302i$
Analytic cond. $73.0761$
Root an. cond. $73.0761$
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  + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)7-s + i·9-s + (0.707 + 0.707i)11-s − i·13-s − i·19-s + 21-s + (−0.707 + 0.707i)23-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (−0.707 + 0.707i)31-s − i·33-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)39-s + (0.707 + 0.707i)41-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)7-s + i·9-s + (0.707 + 0.707i)11-s − i·13-s − i·19-s + 21-s + (−0.707 + 0.707i)23-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (−0.707 + 0.707i)31-s − i·33-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)39-s + (0.707 + 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.953 + 0.302i$
Analytic conductor: \(73.0761\)
Root analytic conductor: \(73.0761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (1:\ ),\ -0.953 + 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03426632157 + 0.2209050246i\)
\(L(\frac12)\) \(\approx\) \(0.03426632157 + 0.2209050246i\)
\(L(1)\) \(\approx\) \(0.7099522467 + 0.01623142691i\)
\(L(1)\) \(\approx\) \(0.7099522467 + 0.01623142691i\)

Euler product

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

−22.14514198755540909190915905872, −21.57087400289541021190154293110, −20.59037832298917741362852500698, −19.73908634645435060000776079205, −18.98657290363417934744394395905, −17.854397024827161967392186683505, −17.08753795330892922092618991473, −16.25352884782411762306518457923, −16.01305338961113991686886537302, −14.63919883185978154395010308098, −13.992497926737379422597032717765, −12.905514307423080478113931038659, −11.97325166190607713188084158154, −11.113043520148562614298403541966, −10.49869907158603312406430543362, −9.376484211349463164580989370574, −8.96153090324453154219119823569, −7.360606141353378963479675968888, −6.479223961162293779250948738281, −5.85320458137934493123627598722, −4.43937065646658112976977719333, −3.99077669173670454298982824771, −2.81267940806588051268792133405, −1.08015677902388982830918980927, −0.06804076060226762280998747876, 1.2696930432864205519092844750, 2.31439095632499629288469008892, 3.50159221940731017932326749312, 4.81838682464934232043731988820, 5.86901737973030584175266251923, 6.35312631574893731118509172927, 7.46969480969627938354652449615, 8.24582855453279320008830134623, 9.51499850481743386414385755690, 10.227935789847356091274773214706, 11.34830116894392833228081054168, 12.273423924840633979849782719460, 12.5891067160521637857152917347, 13.59236505063064998581077643180, 14.62280655499156825181057304269, 15.61399511849642016142435081840, 16.3548781707184167907918237981, 17.32082379510224969785151442326, 17.95814181727144689746879004479, 18.72312809422795600917003598449, 19.57128899935035799161018228924, 20.208966127600231176641945192091, 21.54312404618190767616136513983, 22.262633663323655802935090957910, 22.89510524499516370593460239979

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