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Properties

Label 1-740-740.43-r1-0-0
Degree $1$
Conductor $740$
Sign $-0.988 - 0.148i$
Analytic cond. $79.5240$
Root an. cond. $79.5240$
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 + i·7-s − 9-s + 11-s − 13-s + 17-s + i·19-s − 21-s + 23-s − i·27-s + i·29-s + i·31-s + i·33-s − i·39-s − 41-s + ⋯
L(s)  = 1  + i·3-s + i·7-s − 9-s + 11-s − 13-s + 17-s + i·19-s − 21-s + 23-s − i·27-s + i·29-s + i·31-s + i·33-s − i·39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(79.5240\)
Root analytic conductor: \(79.5240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 740,\ (1:\ ),\ -0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1120803925 + 1.499131158i\)
\(L(\frac12)\) \(\approx\) \(-0.1120803925 + 1.499131158i\)
\(L(1)\) \(\approx\) \(0.8449369730 + 0.5941933785i\)
\(L(1)\) \(\approx\) \(0.8449369730 + 0.5941933785i\)

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 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + T \)
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

−22.08096244932013913755742591011, −20.87111314471836546747659466662, −20.068312614906752998655590152665, −19.357552570226933761471118528815, −18.87309804130248876691211767648, −17.53691681152068273776539206738, −17.19353146068479221207533403936, −16.51876677897857674094247338373, −15.01785450857239328167945335639, −14.39290396123406606499323064484, −13.55499255806185664315613194646, −12.8902465376128137559539909903, −11.86734788531659039783152395247, −11.32245683794823164531230194807, −10.1200214390597804513617447609, −9.25370860239648972315461230891, −8.163089642182598414982297259, −7.235853075887480810382473611297, −6.83518528053911272263253504565, −5.68664418961849509301350604802, −4.57909364074746036323800640367, −3.451246214291847415930509428234, −2.364766734538010662175305438164, −1.17039870108390695481332124644, −0.37132901116501828285734675346, 1.41355424003937333577206220836, 2.793638084679504348389025939233, 3.53580153892092372919932111530, 4.73025195048452004963062333808, 5.43179447593760368575207717938, 6.337530967180191402494059313776, 7.58833760200925073951856026243, 8.726274269484627934072115875763, 9.29179880019005010830775140388, 10.09662808333633405580122244203, 11.01922017406355517348129772036, 12.10354975641396418516119079641, 12.39262887608183060228678124850, 14.085926601881266117263976785502, 14.630271854515294060713623059981, 15.234666957134883086715332265, 16.28083893778206818432329536087, 16.84549586220580662760683984440, 17.67865374915843596750012278008, 18.831877528853778115609898662577, 19.477482372192116231108055509569, 20.41160748762880996561482148248, 21.25977398138091182369890238419, 21.85587044701500452074294285920, 22.51015040574524961711972647015

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