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Properties

Label 1-2960-2960.549-r1-0-0
Degree $1$
Conductor $2960$
Sign $-0.889 - 0.457i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  − i·3-s + 7-s − 9-s − i·11-s + 13-s + i·17-s − 19-s − i·21-s − i·23-s + i·27-s + 29-s + i·31-s − 33-s − i·39-s + 41-s + ⋯
L(s)  = 1  − i·3-s + 7-s − 9-s − i·11-s + 13-s + i·17-s − 19-s − i·21-s − i·23-s + i·27-s + 29-s + i·31-s − 33-s − i·39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.889 - 0.457i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ -0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4738486510 - 1.956327738i\)
\(L(\frac12)\) \(\approx\) \(0.4738486510 - 1.956327738i\)
\(L(1)\) \(\approx\) \(1.038883797 - 0.5044311446i\)
\(L(1)\) \(\approx\) \(1.038883797 - 0.5044311446i\)

Euler product

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

−19.3337922802847727600587341313, −18.33824303446175458794356705561, −17.6696514056530685604477602763, −17.24476751817438438633497100383, −16.272845066659922947264785188991, −15.70940764836067442181610908571, −15.00211182874391324816728634576, −14.51389440411371482648991114294, −13.71698027711270836436917718187, −12.962951077156790545294994523445, −11.73779752393337023031159942335, −11.49870186619672367614934598220, −10.63879792722065765701298478599, −9.987552666334586770823397481714, −9.233464995594666569999582342110, −8.514153136719417939285944705356, −7.85302558965005668322065981196, −6.91876705340730053855808353378, −5.92586830591328753335489154068, −5.178938785818950697800079243400, −4.45121461316240881926332150671, −3.963540742432810933622329670924, −2.85648572176814135987794695274, −2.02334042643077321190251946297, −0.98026049669657723925558522679, 0.34333169138319905167910173582, 1.23637809543812027638302466647, 1.872647479803319110344253087585, 2.8389557382704985501235870465, 3.74675954747094243353593420348, 4.7034497513279595231406976810, 5.65733450933943974907492073473, 6.31924607277976801769950860509, 6.87490180737009217652627542263, 8.148592472583947816208115320811, 8.3397561884462829734328026334, 8.834598680042227387247956461459, 10.37373999994298877558466027055, 10.9049105831972144013554411383, 11.51238037683643330709626952491, 12.29234693662534105703425596205, 13.05931513713414427353978167666, 13.59188882749944183771945066827, 14.508942893083780640897538137618, 14.7213357157623774191401649692, 15.97295322124407456446592860772, 16.59973066531469382483341278856, 17.47147609137118510409895136439, 17.89762658201133104266481291887, 18.6295220027003705688737495176

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