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Properties

Label 1-275-275.239-r1-0-0
Degree $1$
Conductor $275$
Sign $0.193 - 0.981i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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  + (−0.809 − 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)8-s + 9-s + (−0.309 − 0.951i)12-s + 13-s + (0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)18-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)8-s + 9-s + (−0.309 − 0.951i)12-s + 13-s + (0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)18-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.193 - 0.981i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.193 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4599876755 - 0.3779458539i\)
\(L(\frac12)\) \(\approx\) \(0.4599876755 - 0.3779458539i\)
\(L(1)\) \(\approx\) \(0.4823727240 - 0.1433279476i\)
\(L(1)\) \(\approx\) \(0.4823727240 - 0.1433279476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 - T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 - T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−25.917051895698007329325176190930, −24.63293404847156779638882086951, −24.03660175024723463770367817261, −22.99544088391417279115197357407, −22.31783285505257948789724036371, −21.18798147112508102095640006418, −19.87841436506860037038218874353, −18.9397682083453161047271358706, −18.18622970035783663756171923930, −17.42925357034689216848149776443, −16.371309789194958709471281294469, −15.8262950749089127859479448519, −14.97750274856638003677849618593, −13.42489831084783502644642361326, −12.46004471176322938525619508347, −11.168754645663094886151598769285, −10.5770024757671906758513690713, −9.37601006699567840090244708866, −8.56843483532455409541620951786, −7.09494618773794976846344462593, −6.29709784367162087204112351023, −5.61412655125342748871246296740, −4.228270605879432080344448627999, −2.25551308949928510044727374128, −0.71874543323237798864196600085, 0.44710290528854308381965121256, 1.64148408167213046567795113890, 3.36413539817417189406215099416, 4.35258218431324540431859640562, 6.059967721106622555551040920833, 6.870176180343218824925673006983, 7.979997156981153233765309040684, 9.35555582588517872035799826328, 10.17183280834542736117658468973, 11.01307386559298392766690442148, 11.83083068444045577934410659603, 12.88088760772947228393231046685, 13.61305344292473448579636637055, 15.6795536326638198995923183152, 16.19682838010685931499764578481, 17.11860661568034066387527641078, 17.90968931379630691660471878893, 18.769269022129195393632232791298, 19.6381401756618810140061279078, 20.70815772412660695403738083036, 21.50797186299041096316323106950, 22.63356251846681191305368815835, 23.11026446936948424990697283821, 24.41383529050711277239622605342, 25.48676517255923886455777865455

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