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Properties

Label 1-35e2-1225.223-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.0282 - 0.999i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (−0.834 + 0.550i)2-s + (0.880 − 0.473i)3-s + (0.393 − 0.919i)4-s + (−0.473 + 0.880i)6-s + (0.178 + 0.983i)8-s + (0.550 − 0.834i)9-s + (−0.550 − 0.834i)11-s + (−0.0896 − 0.995i)12-s + (0.351 − 0.936i)13-s + (−0.691 − 0.722i)16-s + (0.919 − 0.393i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (0.919 + 0.393i)22-s + (0.0896 − 0.995i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (−0.834 + 0.550i)2-s + (0.880 − 0.473i)3-s + (0.393 − 0.919i)4-s + (−0.473 + 0.880i)6-s + (0.178 + 0.983i)8-s + (0.550 − 0.834i)9-s + (−0.550 − 0.834i)11-s + (−0.0896 − 0.995i)12-s + (0.351 − 0.936i)13-s + (−0.691 − 0.722i)16-s + (0.919 − 0.393i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (0.919 + 0.393i)22-s + (0.0896 − 0.995i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0282 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0282 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.0282 - 0.999i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.0282 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8261105776 - 0.8497503479i\)
\(L(\frac12)\) \(\approx\) \(0.8261105776 - 0.8497503479i\)
\(L(1)\) \(\approx\) \(0.9168919385 - 0.1977376455i\)
\(L(1)\) \(\approx\) \(0.9168919385 - 0.1977376455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.834 + 0.550i)T \)
3 \( 1 + (0.880 - 0.473i)T \)
11 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (0.351 - 0.936i)T \)
17 \( 1 + (0.919 - 0.393i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.0896 - 0.995i)T \)
29 \( 1 + (-0.753 + 0.657i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.0896 + 0.995i)T \)
41 \( 1 + (-0.134 + 0.990i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (-0.266 - 0.963i)T \)
53 \( 1 + (-0.919 - 0.393i)T \)
59 \( 1 + (-0.691 - 0.722i)T \)
61 \( 1 + (-0.858 - 0.512i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.351 - 0.936i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.266 + 0.963i)T \)
89 \( 1 + (0.936 - 0.351i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−21.30140789736715700935793804884, −20.580692133115026188657903176680, −19.660824214775805088644735593582, −19.19982617907816685653976438897, −18.5025996563978443323629969017, −17.530024508249105333520500343565, −16.82323417632973183354969253936, −15.906707347313244145905518115519, −15.36562087496078585454891667712, −14.42545045303539614637408207490, −13.45302937091963316328382776066, −12.77439163470264793225712136357, −11.85691914194422701035137813425, −10.87092762581641416914518699153, −10.25050886862797115639019034797, −9.38643980367967309757503545285, −8.95900011407404393816535717380, −7.85154411956700446358740067843, −7.46761804346589237156335475174, −6.30423001250175958636223336127, −4.80770412127361780201136426317, −4.0080840304875343593031497, −3.14802555466015179422528900905, −2.19233111344529790327742279008, −1.48981729858269355543495001231, 0.55133196173282748833895428940, 1.58190994914144621920856933629, 2.68184598736979925072832512970, 3.45816525691120610704160902675, 4.93375580748893629156251704160, 5.952385314291742415336494295141, 6.62852932228207920615047476516, 7.72281463181824879907320713276, 8.193023163996698667024116769779, 8.75459530239604449825868908818, 9.85785690536116437437587281409, 10.402644096407912594225531957206, 11.40065608642080892646315879336, 12.5106314196490607498720261070, 13.33570833620041914421451987951, 14.13204850938723032923022589524, 14.87885710427024645075073832083, 15.47995119347625194295883774072, 16.402707616470707427818569244432, 17.061458969843667254687857106151, 18.165480914177436745287333608853, 18.68740174146378392488057683998, 19.0506598628022017189573280344, 20.20533045469017245997651052872, 20.5541210382792243177844894546

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