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Properties

Label 1-6025-6025.4922-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.495 + 0.868i$
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  + (0.156 + 0.987i)2-s + (0.156 + 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.0784 + 0.996i)7-s + (−0.453 − 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.852 − 0.522i)11-s + (−0.453 − 0.891i)12-s + (−0.0784 − 0.996i)13-s + (−0.996 + 0.0784i)14-s + (0.809 − 0.587i)16-s + (−0.923 + 0.382i)17-s + (−0.453 − 0.891i)18-s + (−0.649 + 0.760i)19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.156 + 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.0784 + 0.996i)7-s + (−0.453 − 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.852 − 0.522i)11-s + (−0.453 − 0.891i)12-s + (−0.0784 − 0.996i)13-s + (−0.996 + 0.0784i)14-s + (0.809 − 0.587i)16-s + (−0.923 + 0.382i)17-s + (−0.453 − 0.891i)18-s + (−0.649 + 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4922, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8012074756 + 0.4656167898i\)
\(L(\frac12)\) \(\approx\) \(0.8012074756 + 0.4656167898i\)
\(L(1)\) \(\approx\) \(0.6179904998 + 0.6620585511i\)
\(L(1)\) \(\approx\) \(0.6179904998 + 0.6620585511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (0.156 + 0.987i)T \)
7 \( 1 + (-0.0784 + 0.996i)T \)
11 \( 1 + (0.852 - 0.522i)T \)
13 \( 1 + (-0.0784 - 0.996i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (-0.649 + 0.760i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (-0.996 + 0.0784i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.0784 + 0.996i)T \)
79 \( 1 + (-0.453 - 0.891i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.233 - 0.972i)T \)
97 \( 1 + T \)
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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

−17.58580507120312600224756873158, −17.19738919399644284226409869719, −16.701710410723224816214362342709, −15.40586506300433520260668117762, −14.623423558623298687916666032949, −14.03814700559119596943052718358, −13.64666754964357954132128452936, −12.88901382896186180939211239006, −12.44678333827757053589784311478, −11.652293950918520470089492480826, −11.092129714642388264170683397112, −10.591692285794503326493751035101, −9.48387199102744807927864563501, −9.03560000716836739897934450486, −8.480186656986833131973921910995, −7.36733412482460339559280483123, −6.77039293056797966263058973145, −6.41973289232718636814938073884, −4.99245760712798367522334119455, −4.62635838521108221416863034725, −3.66238831333248075294801233330, −3.10059152193995738976802172587, −1.97773164018640697382124185856, −1.68550778851966547433481654489, −0.74638186572024056509728964290, 0.27909876470766679827176118059, 1.84012964596082945181099782261, 2.88768017490587141030377771728, 3.61790530651371797257576820414, 4.13311573779693029045426841732, 5.078129059402441944850456260130, 5.63738025704585924748830059807, 6.10483004420385134977486601725, 6.93864116018883874210552984153, 7.95805757211940409533029112581, 8.686108347901867785031121241484, 8.819553867403251993934342682371, 9.71906117146865102687354184805, 10.2816253069946955518014093596, 11.289395105343344415126608391071, 11.80942261869778449914796888351, 12.858462930546311590923445277730, 13.242532841247239035962693434124, 14.2202043590022679560323550740, 14.78879250197042445206694550117, 15.30707495978313983868589084176, 15.59964047129855801899552224021, 16.46503414894653234255719751611, 17.1470053304579905388495285914, 17.36389458867084117984381455805

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