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Properties

Label 1-245-245.219-r0-0-0
Degree $1$
Conductor $245$
Sign $0.284 + 0.958i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
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  + (0.988 + 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.900 + 0.433i)6-s + (0.900 + 0.433i)8-s + (0.365 − 0.930i)9-s + (0.365 + 0.930i)11-s + (−0.955 + 0.294i)12-s + (−0.623 + 0.781i)13-s + (0.826 + 0.563i)16-s + (0.733 + 0.680i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.222 + 0.974i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)24-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.900 + 0.433i)6-s + (0.900 + 0.433i)8-s + (0.365 − 0.930i)9-s + (0.365 + 0.930i)11-s + (−0.955 + 0.294i)12-s + (−0.623 + 0.781i)13-s + (0.826 + 0.563i)16-s + (0.733 + 0.680i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.222 + 0.974i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.392757570 + 1.039441989i\)
\(L(\frac12)\) \(\approx\) \(1.392757570 + 1.039441989i\)
\(L(1)\) \(\approx\) \(1.396562284 + 0.5565195303i\)
\(L(1)\) \(\approx\) \(1.396562284 + 0.5565195303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.988 + 0.149i)T \)
3 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.955 + 0.294i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 - T \)
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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

−25.38448408770422888821657627641, −24.77831763888864836971582261815, −23.97426613788198587362934689705, −23.045537146917343788167135730801, −22.4464620672624856974948685866, −21.549665095571576349881986787356, −20.59732938372148284852702332095, −19.32320372967030310184051129357, −18.755366632205478789599162924079, −17.255862066577981667419758383748, −16.58968220387452002499152120767, −15.53550615774233825508197423854, −14.37808319784606421560753387021, −13.45379911938048278767056760963, −12.57337217699195264081417798750, −11.742597270045603538367691578650, −10.9395339769405897958355629183, −9.883541087247497853451649350252, −7.96264786367329012428631726229, −7.050583049651842516453440402370, −5.8518675506816542708365956637, −5.31659546689369461408242102170, −3.891726686738521607061059221120, −2.56646127255871009910661670932, −1.093585672910044387080976611294, 1.807353282692063659076435098132, 3.445682740021048914371359787243, 4.55285247667867646131108027199, 5.21505264858712073853306998975, 6.55580613881300208613353254190, 7.16761528760967873148014303433, 8.971908364338792876408400669648, 10.26228882804123156221099873195, 11.09896384250085585334601287108, 12.23662837138687130764973691274, 12.65871772783897364388390544765, 14.256968299442165424378000480441, 14.95789644433550049476973679868, 15.856843615085853831533807844, 16.92056367888473376474157695487, 17.38936026472845941699689438786, 18.9828004081985420117928835287, 20.16850713287332049326954992695, 21.07359220849627994254461626042, 21.90129211203050336092943654248, 22.514282992074114694821768679098, 23.55788924036531098101949877806, 24.0137052177349854790645780027, 25.33334244199793153328697892465, 26.13520048638552265336359504915

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