↑

Properties

Label 1-1520-1520.357-r0-0-0
Degree $1$
Conductor $1520$
Sign $0.999 + 0.0404i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
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  + (−0.939 + 0.342i)3-s + (−0.866 + 0.5i)7-s + (0.766 − 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.939 − 0.342i)13-s + (−0.642 + 0.766i)17-s + (0.642 − 0.766i)21-s + (−0.984 + 0.173i)23-s + (−0.5 + 0.866i)27-s + (−0.642 − 0.766i)29-s + (0.5 + 0.866i)31-s + (0.984 + 0.173i)33-s + 37-s + 39-s + (−0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.866 + 0.5i)7-s + (0.766 − 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.939 − 0.342i)13-s + (−0.642 + 0.766i)17-s + (0.642 − 0.766i)21-s + (−0.984 + 0.173i)23-s + (−0.5 + 0.866i)27-s + (−0.642 − 0.766i)29-s + (0.5 + 0.866i)31-s + (0.984 + 0.173i)33-s + 37-s + 39-s + (−0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0404i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ 0.999 + 0.0404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5091690206 + 0.01030400426i\)
\(L(\frac12)\) \(\approx\) \(0.5091690206 + 0.01030400426i\)
\(L(1)\) \(\approx\) \(0.5701301457 + 0.06272638523i\)
\(L(1)\) \(\approx\) \(0.5701301457 + 0.06272638523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.342 - 0.939i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.642 - 0.766i)T \)
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

−20.34966748433268669171834968195, −19.97061116753208165167683229775, −18.835145603771986358508372327944, −18.438268223063336339909693603533, −17.504447785808553062021561359891, −16.918402503618986778884417087101, −16.12904801490305786874445428634, −15.63470049834494323466571032350, −14.53808950338989608567050564199, −13.47699040371074113962480522655, −13.02218434664856099335015152892, −12.22321957557984137421219192646, −11.541722780716338584425609922331, −10.58443307282762574781711117032, −9.979076234262706817391823298346, −9.29729988877457276246335893057, −7.8586583235697958960662217090, −7.26056272633357244416140619058, −6.58707632599783777139711751565, −5.721580530785359001244251683050, −4.80668026506186184614730453922, −4.14888046211428716149469213656, −2.7759119014188690989888783022, −1.95549263512186186121951636566, −0.51978528755022841317813087915, 0.41906211777786995012820191417, 2.01425870622527171053790108590, 3.005924691887020095839940780597, 3.985749496879290686322738308324, 4.93664386381526048307920889047, 5.76505547049962464725012473933, 6.273187820578433439236945615304, 7.25396712034742020342605557090, 8.213922570817135312160442136264, 9.22902985220907331397213042313, 10.05709469977903745451757783842, 10.50565088578015526989642709627, 11.504585254559914026657558400459, 12.22850238493959034004689731149, 12.88080076688707538003431388901, 13.57293751490211182520849746165, 14.88337322741190158811753189815, 15.497483458852270499914075226405, 16.07890477587248312457550483018, 16.82191577781751843171709522102, 17.57909870368550184671645821186, 18.27256543884572126595926417402, 19.05614963943449592417768858433, 19.75539873929441502575489430334, 20.74147505222438530328583658594

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