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Properties

Label 1-6025-6025.4869-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.996 - 0.0806i$
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  + (−0.994 + 0.104i)2-s + (0.406 − 0.913i)3-s + (0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.998 − 0.0523i)7-s + (−0.951 + 0.309i)8-s + (−0.669 − 0.743i)9-s + (−0.777 − 0.629i)11-s + (0.207 − 0.978i)12-s + (−0.998 + 0.0523i)13-s + (0.998 − 0.0523i)14-s + (0.913 − 0.406i)16-s + (−0.707 − 0.707i)17-s + (0.743 + 0.669i)18-s + (0.838 − 0.544i)19-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.406 − 0.913i)3-s + (0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.998 − 0.0523i)7-s + (−0.951 + 0.309i)8-s + (−0.669 − 0.743i)9-s + (−0.777 − 0.629i)11-s + (0.207 − 0.978i)12-s + (−0.998 + 0.0523i)13-s + (0.998 − 0.0523i)14-s + (0.913 − 0.406i)16-s + (−0.707 − 0.707i)17-s + (0.743 + 0.669i)18-s + (0.838 − 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.996 - 0.0806i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.996 - 0.0806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2919297781 + 0.01178778669i\)
\(L(\frac12)\) \(\approx\) \(0.2919297781 + 0.01178778669i\)
\(L(1)\) \(\approx\) \(0.4713333196 - 0.1851510060i\)
\(L(1)\) \(\approx\) \(0.4713333196 - 0.1851510060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (-0.998 - 0.0523i)T \)
11 \( 1 + (-0.777 - 0.629i)T \)
13 \( 1 + (-0.998 + 0.0523i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.838 - 0.544i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.453 + 0.891i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 - iT \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.965 + 0.258i)T \)
73 \( 1 + (0.453 + 0.891i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.629 - 0.777i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−17.6817240945556715329121045591, −16.94646163727753737844150550072, −16.40712798266645741731604508516, −15.90517324128211747216817199814, −15.12989930655630563674323826547, −14.94179659504117730018304172046, −13.80394226099966502579230728418, −13.072944837262847728369364964520, −12.22351772085923967171402956298, −11.78978399186250949239061511547, −10.59547915183048205460381024379, −10.33100802239329315554975929588, −9.74457544024970051785441600921, −9.287810442950868917606742099374, −8.38600477476187950255476585544, −7.92125929190988617097023256644, −7.09499906218145518404184935029, −6.37415015885602272330128951150, −5.5384577965663750216431938577, −4.72341699147507013408822366111, −3.806031660653347582696515989843, −3.07363193410440708988257812612, −2.4470962889880692791043393392, −1.78726461999239758929481428979, −0.181221623280203125219580399349, 0.46321725868278277578604119221, 1.49355899813837271092744527979, 2.42490626299396566167896496223, 2.89840988012269779338975765432, 3.521239130491330595820259288949, 5.03154383778221649801900364453, 5.761452037981663831404565870868, 6.50415938858127117637278783472, 7.15926096112895256958971491176, 7.544389275536962069420367635510, 8.29786304705324669974200519047, 9.16870975465519593009730352043, 9.43385236887863270076675814474, 10.25562888425622656861521232965, 11.09837204442826187637821513915, 11.710060500718023197950043164766, 12.49261682388087954279213649773, 12.93995482679778994306390908375, 13.90650832202856587560570988122, 14.25088189538130093793030894382, 15.35615148789113391148434426409, 15.91000463379246424955208033530, 16.29570374365231918124884558355, 17.3741257268453460995996332766, 17.68044606364658233003641067140

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