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Properties

Label 1-3015-3015.293-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

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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: $\chi_{3015} (293, \cdot )$
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 \( 1 + (0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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