↑

Properties

Label 1-3015-3015.1667-r0-0-0
Degree $1$
Conductor $3015$
Sign $-0.481 - 0.876i$
Analytic cond. $14.0016$
Root an. cond. $14.0016$
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  + (0.971 − 0.235i)2-s + (0.888 − 0.458i)4-s + (−0.971 + 0.235i)7-s + (0.755 − 0.654i)8-s + (−0.580 + 0.814i)11-s + (−0.945 + 0.327i)13-s + (−0.888 + 0.458i)14-s + (0.580 − 0.814i)16-s + (−0.540 − 0.841i)17-s + (0.959 − 0.281i)19-s + (−0.371 + 0.928i)22-s + (0.618 − 0.786i)23-s + (−0.841 + 0.540i)26-s + (−0.755 + 0.654i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)2-s + (0.888 − 0.458i)4-s + (−0.971 + 0.235i)7-s + (0.755 − 0.654i)8-s + (−0.580 + 0.814i)11-s + (−0.945 + 0.327i)13-s + (−0.888 + 0.458i)14-s + (0.580 − 0.814i)16-s + (−0.540 − 0.841i)17-s + (0.959 − 0.281i)19-s + (−0.371 + 0.928i)22-s + (0.618 − 0.786i)23-s + (−0.841 + 0.540i)26-s + (−0.755 + 0.654i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3015\)    =    \(3^{2} \cdot 5 \cdot 67\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(14.0016\)
Root analytic conductor: \(14.0016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3015} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3015,\ (0:\ ),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9301399914 - 1.572034918i\)
\(L(\frac12)\) \(\approx\) \(0.9301399914 - 1.572034918i\)
\(L(1)\) \(\approx\) \(1.432183648 - 0.4123253880i\)
\(L(1)\) \(\approx\) \(1.432183648 - 0.4123253880i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.971 - 0.235i)T \)
7 \( 1 + (-0.971 + 0.235i)T \)
11 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (-0.945 + 0.327i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.0475 - 0.998i)T \)
43 \( 1 + (-0.458 + 0.888i)T \)
47 \( 1 + (-0.371 - 0.928i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (0.814 + 0.580i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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

−19.49515389117083580678290926510, −18.78481879390473922572726579012, −17.6262399775232557338209951297, −17.0838273906690314653101521781, −16.11919864857433726015598670184, −15.96096180925131676752073214602, −15.004173813339674849142083938838, −14.41075529242355917243621634207, −13.45222056621887925436826330275, −13.14354733208936552803814670610, −12.43342681980901994160281805041, −11.6672335823065835688567970694, −10.84628970913316496719382079656, −10.21711966672213650801646836467, −9.3433849669354850958625370241, −8.33558889491082839829008586129, −7.54163603840110386900726640500, −6.93516826599243280764577624582, −6.08917704913699612301715524162, −5.46704015800125952087862020766, −4.7394094463059465913380833984, −3.66376158402621068197576074160, −3.16921917364698480638709063869, −2.43973096472201179077182998546, −1.21892875367161561988350219633, 0.367617060262231063771473834309, 1.78094377400402152813644755924, 2.75983335288969217398066272917, 2.958210897348389069199162435190, 4.290935844756128489532006552075, 4.80008113085501185237359904058, 5.550255387724434513469176349121, 6.51151361806875067588923196818, 7.051834949391143178799478482236, 7.7003710019920225359343381619, 9.01744457754384713144546597184, 9.85146742215457961096477696072, 10.14419667665899710621913998891, 11.27675401038578839457027302393, 11.9527678322750436250153388693, 12.45130201772995720573644074913, 13.32679507504431239787834607825, 13.631304146737005780748190161422, 14.68411679111189885235169158379, 15.28539752501096118422295109045, 15.82167728925085169512382237310, 16.53425106176471687715093149215, 17.29665216663103618423357975790, 18.327734468127778643996151392348, 18.98688346580461899001822865660

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