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Properties

Label 1-6025-6025.4988-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.613 + 0.789i$
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  + (0.891 + 0.453i)2-s + (−0.453 − 0.891i)3-s + (0.587 + 0.809i)4-s − i·6-s + (0.760 − 0.649i)7-s + (0.156 + 0.987i)8-s + (−0.587 + 0.809i)9-s + (−0.649 − 0.760i)11-s + (0.453 − 0.891i)12-s + (−0.996 − 0.0784i)13-s + (0.972 − 0.233i)14-s + (−0.309 + 0.951i)16-s + (−0.760 − 0.649i)17-s + (−0.891 + 0.453i)18-s + (0.233 − 0.972i)19-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (−0.453 − 0.891i)3-s + (0.587 + 0.809i)4-s − i·6-s + (0.760 − 0.649i)7-s + (0.156 + 0.987i)8-s + (−0.587 + 0.809i)9-s + (−0.649 − 0.760i)11-s + (0.453 − 0.891i)12-s + (−0.996 − 0.0784i)13-s + (0.972 − 0.233i)14-s + (−0.309 + 0.951i)16-s + (−0.760 − 0.649i)17-s + (−0.891 + 0.453i)18-s + (0.233 − 0.972i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4988, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.613 + 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3055226046 + 0.6238685399i\)
\(L(\frac12)\) \(\approx\) \(0.3055226046 + 0.6238685399i\)
\(L(1)\) \(\approx\) \(1.230055730 + 0.009881712061i\)
\(L(1)\) \(\approx\) \(1.230055730 + 0.009881712061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (-0.453 - 0.891i)T \)
7 \( 1 + (0.760 - 0.649i)T \)
11 \( 1 + (-0.649 - 0.760i)T \)
13 \( 1 + (-0.996 - 0.0784i)T \)
17 \( 1 + (-0.760 - 0.649i)T \)
19 \( 1 + (0.233 - 0.972i)T \)
23 \( 1 + (-0.233 + 0.972i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (-0.649 + 0.760i)T \)
37 \( 1 + (-0.852 - 0.522i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.649 + 0.760i)T \)
47 \( 1 + (0.453 - 0.891i)T \)
53 \( 1 + (0.891 + 0.453i)T \)
59 \( 1 + (-0.987 - 0.156i)T \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (0.453 + 0.891i)T \)
71 \( 1 + (-0.0784 + 0.996i)T \)
73 \( 1 + (-0.996 + 0.0784i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.0784 + 0.996i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−17.35819424907046714236819971670, −16.78765586686138058867717277952, −15.867167068131002792018162198140, −15.39199571790041571013965859512, −14.792465080951995199056931596882, −14.49086924297076802615632338651, −13.56587546333031364386350631909, −12.595849560655173042838764452897, −12.17932030410493792517610588345, −11.667392775105093322619311413632, −10.90521366211906058687282644945, −10.192928739906959160315045690429, −9.942337426343229837364470603475, −8.942435993905587099013543882099, −8.166450282643103138277673081, −7.23076816197708359600311509396, −6.33496928276329698728460966195, −5.73573124069722033390202743926, −4.955529328416172762428948057, −4.63903065868694483051640987579, −3.96240608693090069969946045890, −2.974590168882867596383467401138, −2.26257101799993290186162681050, −1.627936683463556730048916781361, −0.12612662099093112941744622153, 1.0619571330561498106506373292, 2.086450532192391411330164684570, 2.67882530804023396188401983652, 3.529244226461285262985673873729, 4.579601120658666224584264731276, 5.27093525098034497113990905691, 5.421084418983395153341740657848, 6.68868180044741769986140609432, 7.071056319924548406193850888731, 7.57405609569682015931428008606, 8.32487161016325552022842981326, 8.954278068758322066462158301898, 10.386484631955884341071723727131, 10.858984507927540640892000006490, 11.7217553494632741522602383817, 11.85372175211004255811544288850, 12.97118478066111436185204074494, 13.316180763764136224968612678901, 14.029066477844519936392101241556, 14.31869011517589118389832052852, 15.37493390280041706949437027212, 15.94470192913314115393667900769, 16.631662652240090999741450788414, 17.37572629721420641158387954939, 17.70361948774679048966088929177

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