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Properties

Label 1-6025-6025.1211-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.463 - 0.886i$
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  + (−0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + 6-s + (−0.951 − 0.309i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.951 − 0.309i)11-s + (−0.309 + 0.951i)12-s + (0.951 + 0.309i)13-s + (0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)18-s + (0.587 + 0.809i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + 6-s + (−0.951 − 0.309i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.951 − 0.309i)11-s + (−0.309 + 0.951i)12-s + (0.951 + 0.309i)13-s + (0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)18-s + (0.587 + 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.463 - 0.886i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.463 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009768384024 + 0.01613578240i\)
\(L(\frac12)\) \(\approx\) \(0.009768384024 + 0.01613578240i\)
\(L(1)\) \(\approx\) \(0.5313209356 + 0.1214725514i\)
\(L(1)\) \(\approx\) \(0.5313209356 + 0.1214725514i\)

Euler product

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

−18.046548676610655831075786415968, −17.3822714327546589608234170501, −16.570581486321706450371398883810, −16.0055730424892123033006350899, −15.483760613221032598546053789323, −14.805382456505959760292902726665, −13.668207695308403905957426624674, −13.19671053686107229086579195583, −12.69335738775181823976706064896, −11.72205796671115723597120906732, −11.25468618083100675464899135741, −10.59581093752533359799941471427, −10.03577465988324935264779576051, −9.4502324992363019527300693255, −8.78096119865395620659991713874, −8.3297609561998176679541431274, −7.15312598694124139804976713715, −6.42516879965564704177891202445, −5.24476674590496975421332704425, −5.1302870976875697797574034373, −3.93351672626466538397838155116, −3.5445193682214865868019968364, −2.68682986814178522339876969213, −2.16220914398399660408875905583, −0.6541488741393499995893542043, 0.008969907230979970348470912534, 1.18092635150650703967092766708, 1.78169087771573377270342177876, 3.12287382062855961558098292666, 3.66938923449883060141504807630, 4.901558135309103458047543375480, 5.50051364331711862440772558161, 6.17719441415363856178008027781, 6.73422371008581198448652441676, 7.28408118939785742867990866977, 8.01174877783493112852925626190, 8.646282601930666208542229190800, 9.27347297237520223132349079140, 10.19763077132983161466604404833, 10.811817878739667831800101714256, 11.45016096919464838818680162170, 12.63967306197993551399085673968, 12.979884241807923882582821812789, 13.677222063287294471292109085470, 13.94090852294500100669563502041, 15.0400737110113255925456746908, 15.696454054433875105613691004737, 16.36848822751000950667887851083, 16.67420495518122274215359408472, 17.5659752152670848927951360137

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