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Properties

Label 1-65e2-4225.1008-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.180 + 0.983i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (−0.953 + 0.301i)2-s + (−0.840 + 0.541i)3-s + (0.818 − 0.574i)4-s + (0.638 − 0.769i)6-s + (0.996 − 0.0804i)7-s + (−0.607 + 0.794i)8-s + (0.414 − 0.910i)9-s + (−0.910 + 0.414i)11-s + (−0.377 + 0.926i)12-s + (−0.926 + 0.377i)14-s + (0.339 − 0.940i)16-s + (0.923 − 0.384i)17-s + (−0.120 + 0.992i)18-s + (−0.207 + 0.978i)19-s + (−0.794 + 0.607i)21-s + (0.743 − 0.669i)22-s + ⋯
L(s)  = 1  + (−0.953 + 0.301i)2-s + (−0.840 + 0.541i)3-s + (0.818 − 0.574i)4-s + (0.638 − 0.769i)6-s + (0.996 − 0.0804i)7-s + (−0.607 + 0.794i)8-s + (0.414 − 0.910i)9-s + (−0.910 + 0.414i)11-s + (−0.377 + 0.926i)12-s + (−0.926 + 0.377i)14-s + (0.339 − 0.940i)16-s + (0.923 − 0.384i)17-s + (−0.120 + 0.992i)18-s + (−0.207 + 0.978i)19-s + (−0.794 + 0.607i)21-s + (0.743 − 0.669i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1008, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4693708488 + 0.5631671419i\)
\(L(\frac12)\) \(\approx\) \(0.4693708488 + 0.5631671419i\)
\(L(1)\) \(\approx\) \(0.5582115601 + 0.1910532939i\)
\(L(1)\) \(\approx\) \(0.5582115601 + 0.1910532939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.953 + 0.301i)T \)
3 \( 1 + (-0.840 + 0.541i)T \)
7 \( 1 + (0.996 - 0.0804i)T \)
11 \( 1 + (-0.910 + 0.414i)T \)
17 \( 1 + (0.923 - 0.384i)T \)
19 \( 1 + (-0.207 + 0.978i)T \)
23 \( 1 + (0.406 - 0.913i)T \)
29 \( 1 + (-0.594 + 0.804i)T \)
31 \( 1 + (-0.985 - 0.168i)T \)
37 \( 1 + (0.554 - 0.832i)T \)
41 \( 1 + (0.254 + 0.966i)T \)
43 \( 1 + (0.999 + 0.0402i)T \)
47 \( 1 + (-0.981 - 0.192i)T \)
53 \( 1 + (0.285 + 0.958i)T \)
59 \( 1 + (-0.613 - 0.789i)T \)
61 \( 1 + (-0.877 - 0.478i)T \)
67 \( 1 + (0.818 + 0.574i)T \)
71 \( 1 + (0.889 + 0.457i)T \)
73 \( 1 + (0.995 + 0.0965i)T \)
79 \( 1 + (-0.981 - 0.192i)T \)
83 \( 1 + (0.998 + 0.0483i)T \)
89 \( 1 + (-0.406 + 0.913i)T \)
97 \( 1 + (-0.471 - 0.881i)T \)
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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

−18.12804089114928754002521085480, −17.64908651334737294102333611265, −16.98036157620378228647734824880, −16.52133529588863013747785473583, −15.5860241155009598916606800873, −15.1240180029953572814070095945, −13.98465952060851268447884037554, −13.16973638240615929786585722141, −12.60483861868736396182509009252, −11.79621140024792995211715990429, −11.18945919030007279582216507086, −10.85647187179705027451859553483, −10.08763235063340151385184091843, −9.215684218233201702366574315487, −8.35275026259809354891841960840, −7.66715625254482845433510851300, −7.39408562556531468409401562453, −6.343644025102794050759268211991, −5.58456227813760612522675107817, −4.99728870902132090331488592466, −3.88176604199128582787673973649, −2.81108272887712187496821761142, −2.017181710992662980585354666061, −1.31120085196747909019357870593, −0.42757179844554008179590985031, 0.81491408123976610126743901703, 1.626963152460553378672202082658, 2.556717815149366612781188245792, 3.67057096053050068234117908667, 4.71614826217896941778682547036, 5.31468149943772212432621223941, 5.8588079140618947824073817882, 6.78169479650504274573413775065, 7.62596767270776980789010287086, 7.99348344525547121457779580903, 9.03174240994264763178522058850, 9.65081515391688993242544334397, 10.41797704918768004957960761730, 10.874242016808318872574398649748, 11.39623608250700765572118172251, 12.319551877912356406246208704975, 12.77844207931444094070474372574, 14.35091778227523632046988815342, 14.58479804081760292998051485223, 15.3546015199529996033863983122, 16.08286026078160147571304452613, 16.69489927307588971846112347446, 17.06092670812083055390845733802, 18.044710894396938286944674494, 18.30303352147213003645273652505

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