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Properties

Label 1-2960-2960.1123-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.384 - 0.923i$
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  + (0.939 − 0.342i)3-s + (−0.984 − 0.173i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.984 + 0.173i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (0.342 − 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (−0.984 − 0.173i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.984 + 0.173i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (0.342 − 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.384 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.728920598 - 1.819396120i\)
\(L(\frac12)\) \(\approx\) \(2.728920598 - 1.819396120i\)
\(L(1)\) \(\approx\) \(1.415207460 - 0.3377537657i\)
\(L(1)\) \(\approx\) \(1.415207460 - 0.3377537657i\)

Euler product

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

−19.18987217113508271710683677081, −18.6614732371541635380275503313, −17.76741562942250099945897239689, −16.64352345309484687496020610375, −16.08282530016587909555889348121, −15.711399535887485250562186214337, −14.83463434422417559345866548923, −14.11627622466209801785698063523, −13.30448445860781511643347630544, −13.08677107943474592274209310090, −12.014427751919872527850189136003, −11.15550824701043938487027437963, −10.31466357075502688748465766361, −9.59793637800419441014655039085, −9.22483520232993283015461342825, −8.12742872054055290878594372178, −7.80469351808775895986230776602, −6.74180115882668192151865851714, −5.92098043366160516710214672153, −5.16118118528521724189173349629, −4.02099613529326252205139540391, −3.4442291066364093137598406480, −2.81048423296474104794751518270, −1.87624805940047697430595481837, −0.80783167963560119617954038034, 0.577893557282447556252364716020, 1.276057730038159125995380809453, 2.6619597410111190517278816495, 2.97019718908474504111336807005, 3.726825881779191275448236245233, 4.84028931014775053368453061716, 5.63325171185889103034411129588, 6.711016006466793562733601410445, 7.22142952367133543627389222214, 7.94340010375000516098549504610, 8.74692481886597405819020357279, 9.42934850317942515340440404586, 10.197407939877111587131569468768, 10.66438550728087010614992649790, 12.020692365882318889705663084592, 12.57067300312591714137736721420, 13.227748590674290930555734622488, 13.72390808237369995435857310544, 14.55187278708964644720665752478, 15.28871692861606559680456055086, 16.020042394198103739441364555033, 16.35561310284387018124908742937, 17.68004261815744173699091350000, 18.247886285973524554389942734019, 18.73113228381014953433577382466

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