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Properties

Label 1-6025-6025.9-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.850 - 0.526i$
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  + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.309 − 0.951i)6-s + (0.994 + 0.104i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.743 + 0.669i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.743 − 0.669i)14-s + (−0.978 + 0.207i)16-s + (−0.951 + 0.309i)17-s + (−0.104 − 0.994i)18-s + (−0.994 − 0.104i)19-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.309 − 0.951i)6-s + (0.994 + 0.104i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.743 + 0.669i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.743 − 0.669i)14-s + (−0.978 + 0.207i)16-s + (−0.951 + 0.309i)17-s + (−0.104 − 0.994i)18-s + (−0.994 − 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080799816 - 3.797694048i\)
\(L(\frac12)\) \(\approx\) \(1.080799816 - 3.797694048i\)
\(L(1)\) \(\approx\) \(1.598541754 - 1.386491252i\)
\(L(1)\) \(\approx\) \(1.598541754 - 1.386491252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (0.994 + 0.104i)T \)
11 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (0.207 + 0.978i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (-0.743 - 0.669i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−17.83404225104899307676592944979, −17.12461970771637821839223285593, −16.4649142116135676231363782299, −16.05049833655690732949668080314, −14.93630931843064524514377535060, −14.789507613272492276195617249531, −14.239461931688379637583874665593, −13.67223440110513397533135014131, −12.90777815766637028914285407401, −12.27706064924092860907834961220, −11.27642440036212411334970082835, −10.96215660977300133166825274722, −9.8073233606169340331350962263, −9.00679698428373634861574436993, −8.52088424929663403925713059561, −8.07387419063005554584764867774, −7.05446854797209911897784379602, −6.761710804452602063288450976026, −5.69899637092001812417835547051, −4.76202498728000024206267934161, −4.4172294567202320874427551126, −3.86437619085082039846218746994, −2.70388840022694239155693017702, −2.39739698905387558479343952431, −1.22571246743839341873772738762, 0.63985452990448171920675910581, 1.66373882768802098570469060212, 2.13474353744573717736986277232, 2.69077919867698002833406400604, 3.77217277623904639573338069169, 4.287063990728974880661795869, 4.90355271432849394178213289186, 5.80718989454112448496054864021, 6.735376919469520156703759199744, 7.25609372729049072473232118509, 8.16785847679305561011869650424, 8.874376494995974008549624997004, 9.44692407289887042797154353, 10.17953724805441064505831945397, 10.934500642270560941570594759877, 11.652588504593452299264812449290, 12.26746041948551733775400633883, 12.880495005432465343456148929835, 13.41970462504933883521724112653, 14.230214745818126737942968586627, 14.69444132867627981029430482850, 15.18662746153805935187274071647, 15.587138970386335093229767519229, 17.13934811149609285433393390117, 17.506991614835971540598130673400

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