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Properties

Label 1-6025-6025.4123-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.935 - 0.353i$
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  + (0.453 + 0.891i)2-s + (−0.891 − 0.453i)3-s + (−0.587 + 0.809i)4-s − i·6-s + (−0.0784 − 0.996i)7-s + (−0.987 − 0.156i)8-s + (0.587 + 0.809i)9-s + (−0.996 + 0.0784i)11-s + (0.891 − 0.453i)12-s + (0.649 − 0.760i)13-s + (0.852 − 0.522i)14-s + (−0.309 − 0.951i)16-s + (0.0784 − 0.996i)17-s + (−0.453 + 0.891i)18-s + (0.522 − 0.852i)19-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (−0.891 − 0.453i)3-s + (−0.587 + 0.809i)4-s − i·6-s + (−0.0784 − 0.996i)7-s + (−0.987 − 0.156i)8-s + (0.587 + 0.809i)9-s + (−0.996 + 0.0784i)11-s + (0.891 − 0.453i)12-s + (0.649 − 0.760i)13-s + (0.852 − 0.522i)14-s + (−0.309 − 0.951i)16-s + (0.0784 − 0.996i)17-s + (−0.453 + 0.891i)18-s + (0.522 − 0.852i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.935 - 0.353i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.935 - 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140970405 - 0.2086405535i\)
\(L(\frac12)\) \(\approx\) \(1.140970405 - 0.2086405535i\)
\(L(1)\) \(\approx\) \(0.8635037916 + 0.1600034912i\)
\(L(1)\) \(\approx\) \(0.8635037916 + 0.1600034912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.453 + 0.891i)T \)
3 \( 1 + (-0.891 - 0.453i)T \)
7 \( 1 + (-0.0784 - 0.996i)T \)
11 \( 1 + (-0.996 + 0.0784i)T \)
13 \( 1 + (0.649 - 0.760i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (0.522 - 0.852i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (-0.996 - 0.0784i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.996 - 0.0784i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.760 - 0.649i)T \)
73 \( 1 + (0.649 + 0.760i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.760 - 0.649i)T \)
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

−18.04251259360549065665110179795, −17.16110879464917908885940150293, −16.22828143641653027783949311033, −15.85257611643654907738333675533, −15.08634692192804466602304662877, −14.51650028639451606718621209190, −13.701376296743501421852630580836, −12.73882934252246161177123174320, −12.53527138597851172933558468544, −11.7679918302325603839668771917, −11.19883793449709233034238105131, −10.59870486587003779753936416923, −9.972401795421436840991729259184, −9.36056580977767244477135736493, −8.59923965250510672631018972903, −7.85653630411002426774247899959, −6.42867151210442692281771627222, −6.09135122446278986701633073882, −5.43127955657606552333644618738, −4.83561044902507338002439049433, −3.94813100100457202447240612153, −3.490576365827943347463347075467, −2.36124267979117973163884914283, −1.82270447249353567260269046409, −0.716700744183629041089382350826, 0.437909008600572300107115362995, 1.17692402202206461695664128759, 2.590617888557830520759515644722, 3.316629217673171306111644796986, 4.24834518060231871693744051123, 4.94545614633573688896730279619, 5.49506382918233564605040475497, 6.091759794138493933632629126298, 7.04245241295748522201637693236, 7.39798206950910220145551916444, 7.87128661394872099203913828312, 8.81300907165278226551036249473, 9.78710668693080436977917544930, 10.425752696935964873014781515988, 11.21638594345971168067229962186, 11.75141922984629353806981287881, 12.7316415754518718343789172115, 13.17085569412760589093596323370, 13.61064467569584518645723994220, 14.218552213914332338723617589097, 15.34151660989004305083199406010, 15.76377943416479849844126153977, 16.43247086690086269209311499445, 16.81040752808587240867362351580, 17.74281587122922461732471852812

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