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Properties

Label 1-1480-1480.1293-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.796 - 0.604i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (−0.984 + 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (0.342 − 0.939i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (0.342 − 0.939i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.796 - 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5581839172 - 0.1877816577i\)
\(L(\frac12)\) \(\approx\) \(0.5581839172 - 0.1877816577i\)
\(L(1)\) \(\approx\) \(0.6172871391 + 0.006780084011i\)
\(L(1)\) \(\approx\) \(0.6172871391 + 0.006780084011i\)

Euler product

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

−21.10039250944850971743979690929, −19.66441580533875111989593526459, −19.18097512353888517466364324677, −18.48905411335914355080417729643, −17.72796083043088637619183628334, −16.99423516404081241512245839723, −16.184765491411240857238562383371, −15.729265848267042952262543203971, −14.827108434238092555085247940608, −13.760144278541426849423033482516, −12.89343190540302630836319433591, −12.40921102564909448781631287628, −11.59649060227405762747556287364, −10.812911596409537433949414296406, −10.15027282820946118502728615535, −9.12427617334723100062745659143, −8.43592461413189589796567768144, −7.21742090174717648476500108302, −6.53067395259677178546091766488, −5.87746856073402313933320234062, −4.97446549399955804153183054251, −4.27436779776803207918631313725, −2.83378772321074250009742034410, −2.19433005613937356154709715532, −0.65168318928952309989926211937, 0.40857489563223292986717068934, 1.749122418474712074957682417185, 2.87298345773566449446846908695, 4.1654942302146711604995607658, 4.602413412300708677203193893201, 5.61095950299605882057524186246, 6.52588027172543271400336537135, 7.1395496884102249295490486327, 7.90946269167517782638836208884, 9.31430984969073987226135770179, 9.92514087910628063307073773292, 10.62364469478413723098913942604, 11.229279184179666991481438894942, 12.40267477927450673781041639062, 12.77901377442591150969057429531, 13.50775891125250493166661436187, 14.78135992356550798088461627031, 15.37895716865263828769216669723, 16.14789263418982298499338347089, 16.95913571018860666177208272927, 17.507951756200389064804622268295, 18.03257714691628296392331496964, 19.24639816173508600086998693763, 19.69533391716395593479653649779, 20.67361349203391618162603508483

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