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Properties

Label 1-6025-6025.144-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.812 + 0.582i$
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  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.406 + 0.913i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.994 + 0.104i)11-s + (0.669 + 0.743i)12-s + (0.207 + 0.978i)13-s + (0.866 − 0.5i)14-s + (0.913 − 0.406i)16-s + (−0.951 − 0.309i)17-s + (0.913 + 0.406i)18-s + (−0.207 + 0.978i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.406 + 0.913i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.994 + 0.104i)11-s + (0.669 + 0.743i)12-s + (0.207 + 0.978i)13-s + (0.866 − 0.5i)14-s + (0.913 − 0.406i)16-s + (−0.951 − 0.309i)17-s + (0.913 + 0.406i)18-s + (−0.207 + 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.812 + 0.582i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.812 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7503746321 + 0.2411935857i\)
\(L(\frac12)\) \(\approx\) \(0.7503746321 + 0.2411935857i\)
\(L(1)\) \(\approx\) \(0.6647224676 - 0.2755496175i\)
\(L(1)\) \(\approx\) \(0.6647224676 - 0.2755496175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.406 + 0.913i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.207 + 0.978i)T \)
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 \( 1 + (-0.309 - 0.951i)T \)
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 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.994 - 0.104i)T \)
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

−17.48215669932936766029418009342, −17.051381907977860626558102725911, −16.2637397004334943371771639968, −15.65158192644832611599625469577, −15.216797830684810155148766873774, −14.7169454278122189133709486738, −13.6590271667702405649211223080, −13.304439184337868579569910061049, −12.63454132267108043365204918703, −11.392037448563096790390951132802, −10.86196908105506646362522634118, −10.34627221791729666353290605518, −9.681282862803984114086480674007, −8.90294395574210045018100979467, −8.21616145319441362357091798670, −7.59089294814469704030809213542, −6.803788918013008388712145764416, −6.138223107333093026754457756611, −5.24081530327500708976494509609, −4.95627808210798153227824331638, −4.1381828644332604127113376083, −3.52746118926214608263668112876, −2.50302335054377830662626667128, −0.96451054733036656049542893780, −0.312807733754577413392678854192, 0.92791162084941741617070221836, 1.79354105128468464579198073552, 2.38282094436193267014591889856, 2.86388346845275745132472212461, 4.15997755189796284731521979869, 4.82216981172392897566231830196, 5.45177082744770531741782843549, 6.188644466686225741675359819356, 7.058231015308365435076336128368, 7.877100890998864963491672163855, 8.58937668735993702751504649437, 8.947155050347717472993858487076, 9.97214710768085172756169230946, 10.825014074699778259498332878718, 11.13526676879993846815671837928, 11.93380873470449216299375851962, 12.410415881602677507727602886643, 12.9419435200539936216095877434, 13.62279957265003219996250986242, 14.31167909863287397888867243214, 14.9166160512885958309362657810, 16.122528119212032362970565519398, 16.46740149199059877546612074074, 17.57089199707097041478916844903, 17.94902120436506852944329838311

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