Properties

Label 1-1520-1520.349-r0-0-0
Degree $1$
Conductor $1520$
Sign $0.983 + 0.178i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
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  + (0.866 + 0.5i)3-s + 7-s + (0.5 + 0.866i)9-s i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s i·27-s + (0.866 − 0.5i)29-s + 31-s + (0.5 − 0.866i)33-s i·37-s − 39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + 7-s + (0.5 + 0.866i)9-s i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s i·27-s + (0.866 − 0.5i)29-s + 31-s + (0.5 − 0.866i)33-s i·37-s − 39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.983 + 0.178i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ 0.983 + 0.178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.502420222 + 0.2256334409i\)
\(L(\frac12)\) \(\approx\) \(2.502420222 + 0.2256334409i\)
\(L(1)\) \(\approx\) \(1.589416456 + 0.1563483810i\)
\(L(1)\) \(\approx\) \(1.589416456 + 0.1563483810i\)

Euler product

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

−20.43171320462679301132306834959, −19.80759967510297736054185894054, −19.29931419396674203593145884864, −18.1767973336377925534329842958, −17.74562815745963336859019332237, −17.12205386398582154327589338929, −15.85767806802354815807429694038, −14.908656708625260673882193392811, −14.74647953994848183141332413519, −13.86178376670129670826185824316, −13.00040059745188700978278738357, −12.2444121442704173890222033992, −11.709487435150290589948287133441, −10.37038186533581010197651780062, −9.876821631978423919389745501250, −8.85566997400986817501857824148, −8.02871804075434148023435899239, −7.57706706342995807445351505412, −6.75668934886542626422512594518, −5.60339642433489747309944822290, −4.665208973551586821685421042470, −3.85453150414752825050555467279, −2.73177576755021918467059234857, −1.95818613787698954425485408666, −1.14531491006976434647894806111, 0.99208009706483620862391174241, 2.26153511149153070800781651406, 2.84213520360646066239128478832, 3.994292512388236879106906529694, 4.70845968201477482811794487631, 5.43678215007952501382338135481, 6.69529417642373412176964658909, 7.649765952826193662318510331959, 8.32071074541572474931210212474, 8.9025727110074308722050152740, 9.92245553297184470346338661987, 10.475843320072604902875255221836, 11.54289102358258169169848727310, 12.06138071394193960195057508466, 13.34901216495037133821416265029, 14.096007619777790582773346210641, 14.37109589442603167022995941433, 15.28680146751193263220559149551, 16.07437882163590077980728070096, 16.75206157590016706962909922620, 17.58789315898543355517945539122, 18.639977787971275989229759189236, 19.0860142762470444240016636982, 19.95717068903449933929156068573, 20.756784197223308312284854434644

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