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Properties

Label 1-1560-1560.1349-r1-0-0
Degree $1$
Conductor $1560$
Sign $0.0128 + 0.999i$
Analytic cond. $167.645$
Root an. cond. $167.645$
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  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(167.645\)
Root analytic conductor: \(167.645\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1560,\ (1:\ ),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2573871621 + 0.2541074963i\)
\(L(\frac12)\) \(\approx\) \(0.2573871621 + 0.2541074963i\)
\(L(1)\) \(\approx\) \(0.8244997997 - 0.1696675065i\)
\(L(1)\) \(\approx\) \(0.8244997997 - 0.1696675065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−20.197945869912074676101613828507, −19.3268585206915188231364808444, −18.74310885147515845770319333981, −17.97362446235470503410100927363, −17.15439259789530045888700231533, −16.45879534262721508727056114068, −15.5816251986150367726289361613, −14.88672760045012630625378001554, −14.38429729540777473969698530707, −13.092080199149224346697613074343, −12.6455924148774716870303616122, −11.921484152847911616672736798946, −11.06895066439668952718637334349, −10.02630514030942248149368094110, −9.52980306225609360298185775466, −8.54881544377810238555704974125, −7.94104075513197875711130277453, −6.655310371042904558226336333636, −6.26491575393746372104247045985, −5.2588158415494135063599331469, −4.25655700011188235563886453342, −3.49975955450151738817625141177, −2.267258288624909287829144210841, −1.70703646824834529915645483931, −0.08317032654542343354739762069, 0.76063768065820698860033982566, 1.90323562929771791415981301446, 3.13786037743810423003998332045, 3.75721352634566909816611646131, 4.72411399740822956832990375800, 5.68826507406648908617690193798, 6.66098976425324212777069173553, 7.183943159680908460240108043709, 8.17755401308223791116153668489, 9.16495176261973240119189504910, 9.64709312564913478849486714834, 10.8578457585780597869738947906, 11.172815603225678367475018250509, 12.19069623339593779899084030737, 13.23896915135735089303428468559, 13.58393188522600917040926281145, 14.44302551595631963562172667663, 15.29868564694965473014560623311, 16.36413362731323956309842422536, 16.51855452077757081274101503213, 17.59648275567586315878664786661, 18.21007704033958819555585101787, 19.23549652272107037918077785687, 19.79555285572897813270575815222, 20.3286243173377052486482101240

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