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Properties

Label 1-760-760.309-r0-0-0
Degree $1$
Conductor $760$
Sign $-0.0389 + 0.999i$
Analytic cond. $3.52942$
Root an. cond. $3.52942$
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 + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + 37-s + 39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + 37-s + 39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.0389 + 0.999i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ -0.0389 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4189445169 + 0.4355986854i\)
\(L(\frac12)\) \(\approx\) \(0.4189445169 + 0.4355986854i\)
\(L(1)\) \(\approx\) \(0.7138120089 + 0.03253364357i\)
\(L(1)\) \(\approx\) \(0.7138120089 + 0.03253364357i\)

Euler product

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

−22.029405812644790269703528003333, −21.806067993859548473827700017481, −20.69786422490794171921811460431, −19.89016345129147797482552811878, −18.561835352876856304233559517257, −18.340551892822013273818236985066, −17.177188698494145811906679579098, −16.69005650828571748215931382570, −15.706082275402151244248375939562, −15.00745970835347498258512885846, −14.18001259411281159646523152130, −12.91687712752903493122601566526, −12.18247569428399176084809181523, −11.37768798735481950547746404197, −10.851677071788500778758897763545, −9.64381674492088676500301339217, −8.98999282329442055276867157076, −7.882502209918902475578662057823, −6.75061222872399393875279566752, −5.9139170905324931945153775570, −5.11351542254690159102234871852, −4.33924780721361738279263010170, −3.04043727445602493558581582610, −1.846146000515267420682216900839, −0.32798745772166506837026570901, 1.33145531081135487690599249579, 2.11002557813282640239329141693, 3.92901075677870464976041215914, 4.60220855922138322878394534995, 5.50255902826860872108360473883, 6.72538277145960184209964458033, 7.198698334762418253672704149521, 8.101370659076624642772407795742, 9.52807191917902997444979633309, 10.20858247166111056682589326087, 11.204396070361585500726802094536, 11.75725660063485563687173153187, 12.73379561629554483018045340513, 13.43418553016281182526497341397, 14.48896241955085452066424323337, 15.25331091077825937622486090726, 16.40168244887046504056214489668, 17.15578293171455070213446683798, 17.557893583869418138042656157654, 18.361127342668419303430519950349, 19.601330493676648285141543334929, 19.93105975747921323133980313354, 21.20248366385650714222845950638, 21.89925650525947245415278502401, 22.70913856877221437193096075332

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