↑

Properties

Label 1-3520-3520.1253-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.898 - 0.439i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)13-s + 17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s − 31-s + (0.382 − 0.923i)37-s + (0.707 + 0.707i)39-s + (−0.707 + 0.707i)41-s + (0.923 + 0.382i)43-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)13-s + 17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s − 31-s + (0.382 − 0.923i)37-s + (0.707 + 0.707i)39-s + (−0.707 + 0.707i)41-s + (0.923 + 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.898 - 0.439i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.175135803 - 0.7357372419i\)
\(L(\frac12)\) \(\approx\) \(3.175135803 - 0.7357372419i\)
\(L(1)\) \(\approx\) \(1.750473829 - 0.2840008261i\)
\(L(1)\) \(\approx\) \(1.750473829 - 0.2840008261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 + T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 - T \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 - iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.729365356599319820283452054582, −18.317720253972782959213311433199, −17.49080570155666988506266118852, −16.561218483618985149467495368412, −15.9313741617011565242745314823, −15.16861597307126361698131428735, −14.764692880312027282312703905892, −14.08316698485037378285901339863, −13.36184599011943785765260009342, −12.591792855874225195073419666075, −11.88391395574222045045361971220, −11.02401082231269110143117614269, −10.27694561349681583947052673785, −9.64139269009531726372550366584, −8.85895537741796009148664912732, −8.14104597402778231766183860722, −7.80335265842220636672246613502, −6.82344255413647402114166240989, −5.5951683509909925971564691398, −5.2354943690740881145145686627, −4.262004129977286042185755552062, −3.36663164815996014926437279299, −2.80507643351877132737408367299, −1.900641143069848255036749293144, −1.01808528774907500400917280773, 1.14408691081883707737164076778, 1.438065378766196160433539521786, 2.573689518871896293165069122174, 3.430586927680372678185904475633, 4.03275392852761904835439454216, 4.93005608156103629892958360524, 5.78950115521438481125411978497, 7.050715520404947009104099744960, 7.21678015753069673080386785223, 8.043795487715004716580454421208, 8.8158726225178152122215363803, 9.42829879563273952823641539862, 10.17442249386790985995050651517, 11.11118400445516010639863262396, 11.69798292097240534882344089807, 12.55071592238948428975108936524, 13.390742184893352544206030242360, 13.81717476208598503732734229683, 14.61424282704072274766915956483, 14.84942062007459830473891719976, 16.09816908531057156322522245714, 16.44612026117930447887140488727, 17.48692312018624782243921725882, 18.121321782112072408172052323257, 18.63912709789143429630054942489

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