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Properties

Label 1-6025-6025.78-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.425 - 0.904i$
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 + (0.358 + 0.933i)3-s + (0.743 + 0.669i)4-s + i·6-s + (−0.999 + 0.0261i)7-s + (0.453 + 0.891i)8-s + (−0.743 + 0.669i)9-s + (0.0261 + 0.999i)11-s + (−0.358 + 0.933i)12-s + (−0.284 − 0.958i)13-s + (−0.942 − 0.333i)14-s + (0.104 + 0.994i)16-s + (−0.522 + 0.852i)17-s + (−0.933 + 0.358i)18-s + (−0.333 − 0.942i)19-s + ⋯
L(s)  = 1  + (0.933 + 0.358i)2-s + (0.358 + 0.933i)3-s + (0.743 + 0.669i)4-s + i·6-s + (−0.999 + 0.0261i)7-s + (0.453 + 0.891i)8-s + (−0.743 + 0.669i)9-s + (0.0261 + 0.999i)11-s + (−0.358 + 0.933i)12-s + (−0.284 − 0.958i)13-s + (−0.942 − 0.333i)14-s + (0.104 + 0.994i)16-s + (−0.522 + 0.852i)17-s + (−0.933 + 0.358i)18-s + (−0.333 − 0.942i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1224768473 + 0.07772223647i\)
\(L(\frac12)\) \(\approx\) \(-0.1224768473 + 0.07772223647i\)
\(L(1)\) \(\approx\) \(1.088234724 + 0.8596509828i\)
\(L(1)\) \(\approx\) \(1.088234724 + 0.8596509828i\)

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 \( 1 + (0.358 + 0.933i)T \)
7 \( 1 + (-0.999 + 0.0261i)T \)
11 \( 1 + (0.0261 + 0.999i)T \)
13 \( 1 + (-0.284 - 0.958i)T \)
17 \( 1 + (-0.522 + 0.852i)T \)
19 \( 1 + (-0.333 - 0.942i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (-0.933 - 0.358i)T \)
31 \( 1 + (-0.878 + 0.477i)T \)
37 \( 1 + (-0.824 - 0.566i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.777 + 0.629i)T \)
59 \( 1 + (0.838 - 0.544i)T \)
61 \( 1 + (0.891 - 0.453i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.725 + 0.688i)T \)
73 \( 1 + (0.972 - 0.233i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.725 + 0.688i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−16.866227606812194793283382805449, −16.47933853753337879863400535568, −15.88516962184690018219408539178, −14.88788524603006493587222919079, −14.35447263260934890142652991988, −13.77442257212436847994420186545, −13.26843951153325874634258199311, −12.65611875008176778732235347295, −12.06732735643567774522681623994, −11.45438072627155414990928067368, −10.81129205910872736886320267344, −9.82025334483209386307263450634, −9.25610947832140068561006680195, −8.4687342850667021409955098098, −7.51389934682008894932843331590, −6.867967803424211084297410995328, −6.28232277452270311284533887643, −5.844696730362188193065766119101, −4.881076707653608218407712130613, −3.79454845236046228725879075499, −3.47144524509039102066356071894, −2.51190268953069099494352611691, −2.05496369637502204247284551774, −1.05920473224501299152951461283, −0.02234354891905225394213449969, 1.99866655092020017652337245327, 2.44232128656612947473281890295, 3.450348127601954667530548547079, 3.79905955579603861465377310745, 4.53986743343935371146808851124, 5.41186500183808697384241082209, 5.77212334773739533012560843056, 6.8160856228761005021118338630, 7.34293763214095162734372173258, 8.165738765891313037595775221752, 8.97050389295196039942545661392, 9.61345388569890166355395944708, 10.421402738809639305608991494787, 10.859202769045025707996987471508, 11.76596240884462277937871608459, 12.66393921950042613196753616485, 12.933729350590322797222144860319, 13.70940427660269551375189354508, 14.44273639599827986015946728379, 15.15245296669353125444612186850, 15.56272162392303232748680757494, 15.85655021938881944402037274138, 16.884292287356212792186861021561, 17.270332908837168773287025941781, 17.943930401444826322857917207241

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