Properties

Label 1-6025-6025.1242-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.969 - 0.243i$
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  + (−0.933 + 0.358i)2-s + (0.838 − 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (−0.942 − 0.333i)7-s + (−0.453 + 0.891i)8-s + (0.406 − 0.913i)9-s + (0.999 + 0.0261i)11-s + (0.258 − 0.965i)12-s + (−0.608 − 0.793i)13-s + (0.999 − 0.0261i)14-s + (0.104 − 0.994i)16-s + (0.233 + 0.972i)17-s + (−0.0523 + 0.998i)18-s + (−0.942 − 0.333i)19-s + ⋯
L(s)  = 1  + (−0.933 + 0.358i)2-s + (0.838 − 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (−0.942 − 0.333i)7-s + (−0.453 + 0.891i)8-s + (0.406 − 0.913i)9-s + (0.999 + 0.0261i)11-s + (0.258 − 0.965i)12-s + (−0.608 − 0.793i)13-s + (0.999 − 0.0261i)14-s + (0.104 − 0.994i)16-s + (0.233 + 0.972i)17-s + (−0.0523 + 0.998i)18-s + (−0.942 − 0.333i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300418293 - 0.1610692160i\)
\(L(\frac12)\) \(\approx\) \(1.300418293 - 0.1610692160i\)
\(L(1)\) \(\approx\) \(0.8621720608 - 0.08784720953i\)
\(L(1)\) \(\approx\) \(0.8621720608 - 0.08784720953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.933 + 0.358i)T \)
3 \( 1 + (0.838 - 0.544i)T \)
7 \( 1 + (-0.942 - 0.333i)T \)
11 \( 1 + (0.999 + 0.0261i)T \)
13 \( 1 + (-0.608 - 0.793i)T \)
17 \( 1 + (0.233 + 0.972i)T \)
19 \( 1 + (-0.942 - 0.333i)T \)
23 \( 1 + (-0.996 + 0.0784i)T \)
29 \( 1 + (0.629 + 0.777i)T \)
31 \( 1 + (0.688 + 0.725i)T \)
37 \( 1 + (0.942 + 0.333i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (-0.649 + 0.760i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (-0.358 + 0.933i)T \)
59 \( 1 + (0.998 - 0.0523i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (0.933 - 0.358i)T \)
71 \( 1 + (0.477 - 0.878i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.987 + 0.156i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.688 - 0.725i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−17.73297282063921411394090755522, −16.92137528469873073838664809010, −16.49711081356539095061950249139, −15.9053397973690405235786016544, −15.28859600668180251573086055525, −14.52568881101989607172465229817, −13.87873310906381319025450846492, −13.09591234000861774871716184332, −12.25877808076456296682651714990, −11.78752931145705695968725164258, −11.03530082831118603475052728043, −9.97021568103917335413436170982, −9.73212128012807569778897981304, −9.32465129740818016439915493856, −8.412156464397439800748975857742, −8.07230657342487891313300883048, −6.886028417915101938185491283112, −6.686119469984448032599986929291, −5.635607809062514200189708253111, −4.33818610815103709045459375189, −3.93623538152172300743119766553, −3.09223618118604114234156771023, −2.34321717578063291222178808413, −1.90382128378392785806396991267, −0.58194780809945407530617666521, 0.67155115803903645328536579961, 1.441840569983438271081168650889, 2.25657879273370884766083419303, 3.05750530192787291335473648168, 3.702478180105082265215843325137, 4.708836604073890814920247734531, 5.93637137825702786276785875866, 6.54802003357689588929469774937, 6.83534008257753803478408353300, 7.79119965879443366009775976997, 8.31644356307767384806846615859, 8.884370343666532862085295888766, 9.72471388828914900116254413837, 10.074200911000246652840253194804, 10.772410003750062392158961766455, 11.94225661364000175448526574749, 12.336233416869098069726255821314, 13.13501899843625405922220146084, 13.834846122379849870782985938049, 14.69005141658798806216593004052, 14.973182889536238028892187128485, 15.71214521199221405588423992761, 16.54327376470711797458893017253, 17.050863950823119428352661834485, 17.74973041773972455736927008393

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