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Properties

Label 1-6025-6025.4231-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.947 - 0.319i$
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 + (0.104 − 0.994i)3-s + (−0.978 + 0.207i)4-s + 6-s + (−0.994 + 0.104i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.994 + 0.104i)11-s + (0.104 + 0.994i)12-s + (−0.406 + 0.913i)13-s + (−0.207 − 0.978i)14-s + (0.913 − 0.406i)16-s + (0.587 − 0.809i)17-s + (0.104 − 0.994i)18-s + (−0.207 + 0.978i)19-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.104 − 0.994i)3-s + (−0.978 + 0.207i)4-s + 6-s + (−0.994 + 0.104i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.994 + 0.104i)11-s + (0.104 + 0.994i)12-s + (−0.406 + 0.913i)13-s + (−0.207 − 0.978i)14-s + (0.913 − 0.406i)16-s + (0.587 − 0.809i)17-s + (0.104 − 0.994i)18-s + (−0.207 + 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.947 - 0.319i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02339987080 + 0.1426761872i\)
\(L(\frac12)\) \(\approx\) \(0.02339987080 + 0.1426761872i\)
\(L(1)\) \(\approx\) \(0.7104038926 + 0.1797446128i\)
\(L(1)\) \(\approx\) \(0.7104038926 + 0.1797446128i\)

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 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.994 + 0.104i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.207 + 0.978i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (0.406 + 0.913i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 - T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (0.994 - 0.104i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.994 - 0.104i)T \)
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

−17.152172246985986335436172710154, −16.97143435401485663434167383100, −15.89865215912066197753718032847, −15.1297554319361672843255630074, −15.05774799299028398315945838889, −13.82890664839674319666050888294, −13.29612183414972918309017887878, −12.824184088318858036550962672962, −12.042755982426569008587103740121, −11.20193714555964774971528842633, −10.61404081370559655473514813509, −10.13052473926631819543078041921, −9.59541866287193603833940262605, −8.97948337041567769416750310067, −8.13991261217799607191897676650, −7.549546690764465068054749871814, −6.096719948272030477503490165413, −5.718888590364795969785624028471, −4.79103824779269511061975644608, −4.33314123150512274644764199515, −3.32523797888150818473087974525, −2.91583477161921271154590216029, −2.41820600547643962581955040714, −0.955773345814659823054250958385, −0.04723430876722053806180562252, 0.946744307773405607477977935380, 2.09122036132545487339112117224, 3.0161434970792469052912231117, 3.51508812445933127522772607704, 4.73372629677162709917199155791, 5.339099998268496035798069374118, 6.0668530898098036506811922139, 6.742597506504285933286977273674, 7.236776932455255864694107214340, 7.77424662378130805133033306977, 8.67641673522011318422085423420, 9.131834492017246552728698758119, 9.93432355055802605829925159425, 10.66541322781648365543304774678, 11.87687278461980256456557008127, 12.39793418999457258901012825668, 12.90982105250113528534325236149, 13.48712657242317075063094277670, 14.321091779925389022782651692794, 14.49946151950701226621175872559, 15.65741717806141472622157521721, 16.05086667105342177763004012284, 16.8640609694135571961739653663, 17.1720120364791531130988809095, 18.25094144020110214127458938177

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