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Properties

Label 1-6025-6025.54-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.987 - 0.160i$
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  + (0.104 − 0.994i)2-s + (0.5 − 0.866i)3-s + (−0.978 − 0.207i)4-s + (−0.809 − 0.587i)6-s + (−0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)9-s + (−0.104 + 0.994i)11-s + (−0.669 + 0.743i)12-s + (0.978 + 0.207i)13-s + (−0.5 + 0.866i)14-s + (0.913 + 0.406i)16-s + (−0.309 − 0.951i)17-s + (−0.913 + 0.406i)18-s + (−0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (0.5 − 0.866i)3-s + (−0.978 − 0.207i)4-s + (−0.809 − 0.587i)6-s + (−0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)9-s + (−0.104 + 0.994i)11-s + (−0.669 + 0.743i)12-s + (0.978 + 0.207i)13-s + (−0.5 + 0.866i)14-s + (0.913 + 0.406i)16-s + (−0.309 − 0.951i)17-s + (−0.913 + 0.406i)18-s + (−0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.987 - 0.160i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5708597381 - 0.04607765880i\)
\(L(\frac12)\) \(\approx\) \(0.5708597381 - 0.04607765880i\)
\(L(1)\) \(\approx\) \(0.6417276763 - 0.6290626689i\)
\(L(1)\) \(\approx\) \(0.6417276763 - 0.6290626689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.104 - 0.994i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−17.39743394047812439663668161984, −16.86230826961319826927728798426, −16.2968534384507439703140786104, −15.65418064697022254946103981026, −15.284213976711239219906194403, −14.715214713777948934423005019439, −13.812915130212922168567518200003, −13.24483415820799447246599309972, −12.95588637944205362144271826116, −11.81238075554437014107789384651, −10.790584160856830809881201273092, −10.44342259342281300069817417450, −9.37244876846588711349410388612, −9.03880264371853465056512098639, −8.379964846855259269975030148087, −7.92274378137678599428833201454, −6.77815880155244878004715390465, −6.09135926254807326997341884574, −5.72738728654491219204723968544, −4.78315283103009148189522810528, −4.06080402202940945784581121037, −3.301915837394083377012604948974, −2.9890760154701906186176077003, −1.581316062043642790792455554029, −0.152908237247438621697600900250, 0.870547401431697079711730094029, 1.623703683160207266377049913181, 2.55996988019627495674967184147, 2.86590237724417165305501172519, 4.03999171967548315991607512366, 4.22004827560303239066129827203, 5.51308935490660532943137771715, 6.247562410401077294414974157125, 6.97472290565420989176726194733, 7.628181319908543983596677238553, 8.51834698856625869179302180141, 9.2505112973664931108998903610, 9.51960886292599578546552393623, 10.55422818788007969874295019421, 11.07038522017159122028232077051, 11.911444018497959134563331774251, 12.60629424911264454262044026606, 13.04077221028757898570401471812, 13.440282817350269987340067054915, 14.248467697187397809994633675091, 14.79112082544359172663276496630, 15.60910132778796541786036711393, 16.510670269724882090506928119635, 17.34021815267145618355744351345, 17.87817302115966161885279485600

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