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Properties

Label 1-6025-6025.3348-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: $\chi_{6025} (3348, \cdot )$
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

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

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