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Properties

Label 1-6025-6025.5798-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.152 - 0.988i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.0523 − 0.998i)2-s + (−0.358 − 0.933i)3-s + (−0.994 − 0.104i)4-s + (−0.951 + 0.309i)6-s + (−0.958 − 0.284i)7-s + (−0.156 + 0.987i)8-s + (−0.743 + 0.669i)9-s + (−0.333 − 0.942i)11-s + (0.258 + 0.965i)12-s + (0.608 − 0.793i)13-s + (−0.333 + 0.942i)14-s + (0.978 + 0.207i)16-s + (−0.996 + 0.0784i)17-s + (0.629 + 0.777i)18-s + (−0.958 − 0.284i)19-s + ⋯
L(s)  = 1  + (0.0523 − 0.998i)2-s + (−0.358 − 0.933i)3-s + (−0.994 − 0.104i)4-s + (−0.951 + 0.309i)6-s + (−0.958 − 0.284i)7-s + (−0.156 + 0.987i)8-s + (−0.743 + 0.669i)9-s + (−0.333 − 0.942i)11-s + (0.258 + 0.965i)12-s + (0.608 − 0.793i)13-s + (−0.333 + 0.942i)14-s + (0.978 + 0.207i)16-s + (−0.996 + 0.0784i)17-s + (0.629 + 0.777i)18-s + (−0.958 − 0.284i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.152 - 0.988i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5798, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2252897252 - 0.1932250349i\)
\(L(\frac12)\) \(\approx\) \(0.2252897252 - 0.1932250349i\)
\(L(1)\) \(\approx\) \(0.3523968676 - 0.4551264455i\)
\(L(1)\) \(\approx\) \(0.3523968676 - 0.4551264455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.0523 - 0.998i)T \)
3 \( 1 + (-0.358 - 0.933i)T \)
7 \( 1 + (-0.958 - 0.284i)T \)
11 \( 1 + (-0.333 - 0.942i)T \)
13 \( 1 + (0.608 - 0.793i)T \)
17 \( 1 + (-0.996 + 0.0784i)T \)
19 \( 1 + (-0.958 - 0.284i)T \)
23 \( 1 + (-0.852 + 0.522i)T \)
29 \( 1 + (-0.544 - 0.838i)T \)
31 \( 1 + (-0.902 - 0.430i)T \)
37 \( 1 + (0.958 + 0.284i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (-0.972 - 0.233i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (-0.777 - 0.629i)T \)
61 \( 1 + (0.156 + 0.987i)T \)
67 \( 1 + (-0.0523 + 0.998i)T \)
71 \( 1 + (-0.983 + 0.182i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.902 + 0.430i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−17.74373291807683141982819017449, −16.94993157040818102241492986812, −16.3405590936355263038483649366, −16.029713684286148193472618756991, −15.34146073061301921086522632909, −14.80609976044454108468045175945, −14.20047712084725423515180749574, −13.28527309080909884091659675142, −12.67804042527267325141836839315, −12.13729346652369611015391705366, −11.04931050360302697105761858019, −10.45023064255217286786216307654, −9.685596444471584536868080861187, −9.141721905479605309236523968251, −8.72747705241183778980940026970, −7.7654523727994915848446889824, −6.79300825733155898498809787325, −6.38577324138745572359538839937, −5.77739361540166113263585305298, −4.92783833580555038176345267333, −4.2342826802663978092671934083, −3.84376824474385446337418623617, −2.84881186803970139374894169427, −1.79214107224017776668151264149, −0.16046477129917361468497498916, 0.46221946395081033117072721192, 1.357156165678229817582827959285, 2.33621212878083407112545210168, 2.82758373928055790572576174987, 3.714289371439850657485339256450, 4.34296503671796439510196383992, 5.570004405957122727461885997035, 5.92476107009797040227016578044, 6.59273838174222289551314365133, 7.6862094220249265363677046593, 8.19844144333901825179654280712, 8.95484687296787161931016811808, 9.64056185695541770724749405218, 10.6625792016959728171892196660, 10.90429472015571565815746181204, 11.56955227483156870790622116030, 12.413843388157841393664622559081, 13.05264958684514659638243903961, 13.33855410130144915123067725156, 13.78919637742560547415078019215, 14.78493411504836747377156831271, 15.65837096478928599392341056713, 16.42324021383207298857030662825, 17.14377465799796871347965799653, 17.72011476016374463140136671388

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