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Properties

Label 1-6025-6025.3342-r1-0-0
Degree $1$
Conductor $6025$
Sign $0.484 - 0.874i$
Analytic cond. $647.476$
Root an. cond. $647.476$
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 + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.258 − 0.965i)7-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.629 − 0.777i)11-s + (−0.913 + 0.406i)12-s + (0.777 − 0.629i)13-s + (−0.933 + 0.358i)14-s + (0.913 − 0.406i)16-s + (−0.453 + 0.891i)17-s + (−0.5 − 0.866i)18-s + (−0.544 − 0.838i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.258 − 0.965i)7-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.629 − 0.777i)11-s + (−0.913 + 0.406i)12-s + (0.777 − 0.629i)13-s + (−0.933 + 0.358i)14-s + (0.913 − 0.406i)16-s + (−0.453 + 0.891i)17-s + (−0.5 − 0.866i)18-s + (−0.544 − 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(647.476\)
Root analytic conductor: \(647.476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (1:\ ),\ 0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.565322187 - 1.510915177i\)
\(L(\frac12)\) \(\approx\) \(2.565322187 - 1.510915177i\)
\(L(1)\) \(\approx\) \(1.098662031 - 0.7787563410i\)
\(L(1)\) \(\approx\) \(1.098662031 - 0.7787563410i\)

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 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.629 - 0.777i)T \)
13 \( 1 + (0.777 - 0.629i)T \)
17 \( 1 + (-0.453 + 0.891i)T \)
19 \( 1 + (-0.544 - 0.838i)T \)
23 \( 1 + (-0.156 + 0.987i)T \)
29 \( 1 + (0.743 - 0.669i)T \)
31 \( 1 + (-0.838 + 0.544i)T \)
37 \( 1 + (-0.358 + 0.933i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (0.998 + 0.0523i)T \)
73 \( 1 + (-0.156 + 0.987i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (0.777 + 0.629i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−17.82928406862113362939041335404, −16.60018374072938478444707190224, −16.36486800005542554862608392779, −15.56555781720861765036181302875, −15.11072633813577259761314419523, −14.45687334098323402005265887349, −14.01017132585222176297120622057, −13.28499879209630393719011002205, −12.495557717921665055914524029114, −12.05358281329533806543542627554, −10.7889582937244970091769456965, −10.06317425515967701718467013473, −9.34936736642567428596249162036, −8.744438069582167997315073600218, −8.62976375596000106555982468929, −7.53707607048911737525197868178, −6.91870215764065562247610479933, −6.35138017726669265581585221124, −5.48389973921485960997168207440, −4.667903117697685135214465353792, −4.02103798043047816644833189858, −3.40454343312140130016482542069, −2.226142108218309353998584379473, −1.739765310995895945146820750103, −0.43173219527336618483215323991, 0.79601024756816431126719510756, 1.21833799965362539532218606131, 2.06596265498838539721650163636, 3.076300354251677331402531586352, 3.48264471754498232914332559520, 4.09168096047412611074611386381, 4.779766165650997253458941994325, 6.03370750908962624147463789286, 6.66651214478599602593222299108, 7.65977633133774106957376663019, 8.24932201737773634573624497346, 8.80798300541177980076940093121, 9.44229281588902871388252145669, 10.19643138564774646459049245117, 10.776007450810053580290498994815, 11.35069206508184278571531176445, 12.24472782895966190593832744068, 13.16389735419395598689025344565, 13.29169674489559461416247771544, 13.89959325811136573490592088809, 14.59316326041609960969890175792, 15.3401852494836144090000329913, 16.12158720356977180119016171256, 17.00759159233449571030042505119, 17.60894473347751677782865713911

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