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Properties

Label 1-6025-6025.862-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.997 - 0.0731i$
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  + (0.453 − 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + i·6-s + (0.0784 − 0.996i)7-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (0.996 + 0.0784i)11-s + (0.891 + 0.453i)12-s + (−0.649 − 0.760i)13-s + (−0.852 − 0.522i)14-s + (−0.309 + 0.951i)16-s + (−0.0784 − 0.996i)17-s + (−0.453 − 0.891i)18-s + (−0.522 − 0.852i)19-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + i·6-s + (0.0784 − 0.996i)7-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (0.996 + 0.0784i)11-s + (0.891 + 0.453i)12-s + (−0.649 − 0.760i)13-s + (−0.852 − 0.522i)14-s + (−0.309 + 0.951i)16-s + (−0.0784 − 0.996i)17-s + (−0.453 − 0.891i)18-s + (−0.522 − 0.852i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.997 - 0.0731i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (862, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.997 - 0.0731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05755806661 - 1.571656368i\)
\(L(\frac12)\) \(\approx\) \(0.05755806661 - 1.571656368i\)
\(L(1)\) \(\approx\) \(0.7496334987 - 0.6788138734i\)
\(L(1)\) \(\approx\) \(0.7496334987 - 0.6788138734i\)

Euler product

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

−17.707603649865880373776426110, −17.337088577122413604481386758909, −16.6474303582318834875241861907, −16.26753438079519871996533299851, −15.39575409710702555098553253218, −14.72283642596788536740788967058, −14.27899902186627287671992956321, −13.46697373675016928820497769320, −12.42823927695156006827304072160, −12.39093431677078303111911734413, −11.804591452119556607965669421370, −10.89565022100549246719623141677, −10.07683042056617226402542037438, −9.06047783694417360739663620778, −8.64350355283288895046421391240, −7.860700041466829091916227271027, −6.9594464622056195218859337221, −6.515166907295438095710363074245, −5.93099286735503589140061274174, −5.343520252427227485426187766591, −4.44538001706064820251700482600, −4.06754860963451325391381121767, −2.81463095821399732467045476223, −2.00798202113025611921348191767, −1.029007160907685658996750407347, 0.5774849326726661333144401363, 0.8459867321942893335828252540, 2.02719540926594669483315659055, 3.00169631752124134264070492496, 3.76427576894895309242869185641, 4.39445239670250895367357613431, 4.93952837108130290157301154758, 5.58022633860393164407789596800, 6.57424379894246789020229556328, 6.97030384332214619828318351292, 8.011232502427714880121139081836, 9.25259085910189498317083497114, 9.51771600306402470258172788940, 10.29527547054609910142431609947, 10.86914503540011068965268172461, 11.39801615087845608009428723769, 12.04208916628919531598908058398, 12.58744492354767617807616628547, 13.44202526087625439188153495221, 13.9484830164000240756893116630, 14.69837485585058312356728228290, 15.455508143576143950749018684232, 15.91629405326008870319008011426, 17.10114605919165214743554564014, 17.46027944040965395143074569944

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