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Properties

Label 1-6025-6025.121-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.117 - 0.993i$
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  + (0.406 + 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.258 + 0.965i)7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (0.358 − 0.933i)11-s + (−0.994 + 0.104i)12-s + (0.358 + 0.933i)13-s + (−0.777 + 0.629i)14-s + (−0.104 − 0.994i)16-s + (0.891 − 0.453i)17-s + (−0.866 + 0.5i)18-s + (−0.998 − 0.0523i)19-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.258 + 0.965i)7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (0.358 − 0.933i)11-s + (−0.994 + 0.104i)12-s + (0.358 + 0.933i)13-s + (−0.777 + 0.629i)14-s + (−0.104 − 0.994i)16-s + (0.891 − 0.453i)17-s + (−0.866 + 0.5i)18-s + (−0.998 − 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.117 - 0.993i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.028503491 + 1.156976226i\)
\(L(\frac12)\) \(\approx\) \(-1.028503491 + 1.156976226i\)
\(L(1)\) \(\approx\) \(0.7528181845 + 1.133207839i\)
\(L(1)\) \(\approx\) \(0.7528181845 + 1.133207839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (0.743 + 0.669i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (0.358 - 0.933i)T \)
13 \( 1 + (0.358 + 0.933i)T \)
17 \( 1 + (0.891 - 0.453i)T \)
19 \( 1 + (-0.998 - 0.0523i)T \)
23 \( 1 + (-0.987 + 0.156i)T \)
29 \( 1 + (-0.207 + 0.978i)T \)
31 \( 1 + (-0.0523 + 0.998i)T \)
37 \( 1 + (-0.777 - 0.629i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (-0.544 - 0.838i)T \)
73 \( 1 + (-0.987 + 0.156i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.933 + 0.358i)T \)
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

−17.40957020648633120599883956110, −16.88879371642439678670191839100, −15.56450119583378808742049032184, −14.88774397640879190864856947506, −14.57256584064019785321716153787, −13.790606434760096887193978862449, −13.17594894455382788190212811043, −12.81113569064639747006586835494, −12.005108122758343234627981351281, −11.50867601933654843495777230117, −10.39614513900972930493016812454, −10.10948650510060119766551644335, −9.456048417172275206716738912099, −8.34701078778734280973982367839, −8.07369491782599512252384753439, −7.19343722099276440699978639970, −6.33711208155828387844524140578, −5.73672988928559061615671851563, −4.57447849396922131744896885284, −4.02961950386780440541725251132, −3.45827056535727688042749370305, −2.59937627123373993978000200047, −1.77357118592082150762712938286, −1.300073637720170981487931846026, −0.27648535918110507316897627366, 1.54104886330881143641748351302, 2.44819889056864715945284618595, 3.30382366045970973892632454639, 3.83034277151333426064815148503, 4.5546448498705092097866595419, 5.45987326619308382792398244400, 5.76148850096513300306534331853, 6.781982285306731379145372732656, 7.44191453681786319582358140763, 8.41964601296783515765293824464, 8.79453333114754593007086245155, 9.05786327338356183111474385986, 10.10338550637804915202038915831, 10.84480364761072414552218616919, 11.93642558673579140722417220862, 12.14293654451416574693392391172, 13.31841660226608371185651660450, 13.80209172753959687803454179879, 14.499334138687013639165849674169, 14.714166312156895480931896174587, 15.679881387280882916404643765373, 16.08057355894850299026632369797, 16.57050734509414423376432530988, 17.29053459941989522274191572226, 18.242952783508185557859124865645

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