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Properties

Label 1-35e2-1225.447-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.340 + 0.940i$
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  + (−0.919 − 0.393i)2-s + (−0.834 − 0.550i)3-s + (0.691 + 0.722i)4-s + (0.550 + 0.834i)6-s + (−0.351 − 0.936i)8-s + (0.393 + 0.919i)9-s + (−0.393 + 0.919i)11-s + (−0.178 − 0.983i)12-s + (−0.657 + 0.753i)13-s + (−0.0448 + 0.998i)16-s + (0.722 + 0.691i)17-s − i·18-s + (0.309 − 0.951i)19-s + (0.722 − 0.691i)22-s + (0.178 − 0.983i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (−0.919 − 0.393i)2-s + (−0.834 − 0.550i)3-s + (0.691 + 0.722i)4-s + (0.550 + 0.834i)6-s + (−0.351 − 0.936i)8-s + (0.393 + 0.919i)9-s + (−0.393 + 0.919i)11-s + (−0.178 − 0.983i)12-s + (−0.657 + 0.753i)13-s + (−0.0448 + 0.998i)16-s + (0.722 + 0.691i)17-s − i·18-s + (0.309 − 0.951i)19-s + (0.722 − 0.691i)22-s + (0.178 − 0.983i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1669022686 + 0.2379640441i\)
\(L(\frac12)\) \(\approx\) \(0.1669022686 + 0.2379640441i\)
\(L(1)\) \(\approx\) \(0.4742833060 - 0.03980963621i\)
\(L(1)\) \(\approx\) \(0.4742833060 - 0.03980963621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.919 - 0.393i)T \)
3 \( 1 + (-0.834 - 0.550i)T \)
11 \( 1 + (-0.393 + 0.919i)T \)
13 \( 1 + (-0.657 + 0.753i)T \)
17 \( 1 + (0.722 + 0.691i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.178 - 0.983i)T \)
29 \( 1 + (-0.134 + 0.990i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.178 + 0.983i)T \)
41 \( 1 + (0.963 + 0.266i)T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (-0.512 - 0.858i)T \)
53 \( 1 + (-0.722 + 0.691i)T \)
59 \( 1 + (-0.0448 + 0.998i)T \)
61 \( 1 + (-0.473 - 0.880i)T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (0.657 + 0.753i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.512 + 0.858i)T \)
89 \( 1 + (0.753 - 0.657i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−20.93775220590058713401413997875, −20.11089295944506866772844853255, −19.098923030176529214882385875177, −18.510507966261870209226941085402, −17.67391046180239699614411059688, −17.10820505199903499286796105185, −16.25019929231854227716115699001, −15.90603431712799843021446614714, −14.95633382890767047485126386067, −14.26625308564680003609552108240, −13.02624937627726917808468623573, −11.96194071901838097065982646866, −11.32337123241493213505406537512, −10.60991217699419604748893163164, −9.73039429829101147822785286770, −9.38257162685932571715494455495, −7.9690592000828104332748005297, −7.587190103404901773761813854408, −6.34032525701411497402696174513, −5.62168646534318112541932198091, −5.14207456096583434395186190783, −3.704111871600035701240951102181, −2.68298527852382673394241849000, −1.22147095121642697447335138650, −0.20460934796504796375716856583, 1.224371248184385777818677648688, 2.02099681561828435465056304122, 2.98362317913640708647806822879, 4.408956258905208243447620759420, 5.22573620771051593116176058666, 6.52788218355795658657013113979, 7.032076880827472218418939665732, 7.769568666642619073552645791001, 8.72738073558302660044250425396, 9.72470386989906621739978729259, 10.42185749270661035369326525220, 11.11219218216796398066035933378, 12.05773464313861534415764678418, 12.467081270161307884308436034376, 13.232368115531371117380362793018, 14.48026359267317590795673999669, 15.4493194278477997988410594366, 16.38921700763344014433726999875, 16.912402199675536862886114904517, 17.62543442235468109231580254866, 18.30842748746225198145326641169, 18.887621670646353044926459883030, 19.73067972719389474555604092091, 20.34611985591606451840832915429, 21.51256838407729337810260370403

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