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Properties

Label 1-6025-6025.5628-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.226 + 0.973i$
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  + (−0.987 + 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s − i·6-s + (0.233 − 0.972i)7-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (−0.972 − 0.233i)11-s + (0.156 + 0.987i)12-s + (−0.852 + 0.522i)13-s + (−0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (−0.233 − 0.972i)17-s + (0.987 + 0.156i)18-s + (−0.996 + 0.0784i)19-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s − i·6-s + (0.233 − 0.972i)7-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (−0.972 − 0.233i)11-s + (0.156 + 0.987i)12-s + (−0.852 + 0.522i)13-s + (−0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (−0.233 − 0.972i)17-s + (0.987 + 0.156i)18-s + (−0.996 + 0.0784i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.226 + 0.973i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5628, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.226 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2580278272 + 0.2048163394i\)
\(L(\frac12)\) \(\approx\) \(0.2580278272 + 0.2048163394i\)
\(L(1)\) \(\approx\) \(0.4696001923 + 0.09570614374i\)
\(L(1)\) \(\approx\) \(0.4696001923 + 0.09570614374i\)

Euler product

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

−17.62304875427362910207426520853, −17.10587465229781842147232637614, −16.64139299263367959616613713175, −15.51538670102570296922913878657, −15.02388096292425515437861574158, −14.64201675445322340893639260530, −13.14443823404268660938541253339, −12.85641427869924040446565696936, −12.391143835241604884792617685340, −11.435520373567584636508715501516, −11.11082205657861950051136379125, −10.26927948918142166262327988738, −9.54826850530036703803119743135, −8.649538710983601863807925057204, −8.24872454835021556133029829655, −7.65563154924641192145844408312, −6.938924975294886158533848046897, −6.245779894884674375375526442882, −5.53105191749908650145390519673, −4.89131450633370849627039879026, −3.417511100131413862589501431002, −2.65445520510449404780676976911, −2.08133047586777847051378476147, −1.5538054014337491482380143960, −0.22924796471650500776227802681, 0.44599693396865904893570334719, 1.69394326205835022909990532358, 2.60028734056296922184129861608, 3.25869211062517941489608397098, 4.27829273462918110911533166174, 4.95538052107797508949713285397, 5.585133372878494934693353166903, 6.54405419678518746644769674841, 7.29113430725488608552621130602, 7.74544171901696489247366962618, 8.737661693837764681941629274769, 9.19589857220037518927694885127, 9.93237391460329375785945452593, 10.48071151445435634266212866942, 11.09074692548296711630601899999, 11.41171724187393507427488305195, 12.457839912569344643056903355579, 13.28605508814613496277469458718, 14.25963554542392849647817038846, 14.743744575779060504076780077600, 15.35194653557717053560708720108, 16.15382858166282208792200846943, 16.59351307288550727175235847132, 17.06718167709055549229138642820, 17.66614040443617512742851294515

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