↑

Properties

Label 1-385-385.157-r0-0-0
Degree $1$
Conductor $385$
Sign $0.180 + 0.983i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.866 + 0.5i)12-s + (0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (−0.743 − 0.669i)17-s + (0.994 + 0.104i)18-s + (−0.104 − 0.994i)19-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.866 + 0.5i)12-s + (0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (−0.743 − 0.669i)17-s + (0.994 + 0.104i)18-s + (−0.104 − 0.994i)19-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5868700208 + 0.4891143257i\)
\(L(\frac12)\) \(\approx\) \(0.5868700208 + 0.4891143257i\)
\(L(1)\) \(\approx\) \(0.6104430950 + 0.3290113030i\)
\(L(1)\) \(\approx\) \(0.6104430950 + 0.3290113030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.994 - 0.104i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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

−24.61553418579447549857846331010, −23.20450856459533368395155224604, −22.7350877808340924291370172857, −21.515569685052040038984960377142, −20.728220805131001137063740989334, −19.653473864871973258394981741739, −19.08276228140808419051128204346, −18.14872545867627206775994073, −17.59754200965071427012563475965, −16.67699192241811032849342102236, −15.8004370868424238818132625836, −14.29865750551112150919992446352, −13.09230697837668731279138186495, −12.70931594803545381746959064648, −11.53253176332280861002701810761, −10.9333220535476871035179232790, −9.936213110699264503849403247777, −8.51856709085389189586114468699, −8.09572722594966301752043730846, −6.831793254337415909672345849555, −6.02954674061803281165823406344, −4.42950319396492697153079516544, −3.08293623858262937011803133321, −1.94241957919363519123445307789, −0.88262748331281165719848399647, 0.94972440867556607496152757136, 2.768150252274259663528259387229, 4.33369124007303252570289749422, 5.17793658797111473638189902670, 6.26738127304726071102039811209, 7.06594329170282803026764494245, 8.569111840772607068572498534814, 9.09680476133057211995736356072, 10.093853839984762593383283517958, 11.02604928366224994262223072328, 11.61864409373890070499986941640, 13.3590844980493261703238905119, 14.28304578111818579678411527635, 15.48968268706003955256775739997, 15.72538388244326261632950496030, 16.7704344979319742271713980126, 17.53276496428418757916616120046, 18.28138446810287036179472986434, 19.39587824637689827798827447488, 20.30918693252885357383949557244, 21.146805678649414255153045398256, 22.21505559708862838718477743826, 23.123234037789468083639140207410, 23.76649347365170989660340783131, 24.84315571047297979112367922502

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