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Properties

Label 1-245-245.239-r0-0-0
Degree $1$
Conductor $245$
Sign $0.718 + 0.695i$
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  + (0.222 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s − 18-s + 19-s + (0.900 − 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s − 18-s + 19-s + (0.900 − 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9149435004 + 0.3704195704i\)
\(L(\frac12)\) \(\approx\) \(0.9149435004 + 0.3704195704i\)
\(L(1)\) \(\approx\) \(0.8830770625 + 0.2662845175i\)
\(L(1)\) \(\approx\) \(0.8830770625 + 0.2662845175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
3 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−26.33760709108994154954722177406, −25.14513316641174582745553442880, −23.68049716488895803650673564393, −22.84424815211383066842157233842, −22.45081689811299806589016347350, −21.24082748333071216317107399286, −20.63405262076717654151951415707, −19.860044941532280074232625314888, −18.43554187113775425483994637941, −17.79666054241903426827835781871, −16.77845083807598939533239526298, −15.46293488303913084024428241598, −14.76811294311432457386334898947, −13.46826278887583236815086151297, −12.39053247807563706942916988341, −11.65550591776159220924045682280, −10.571785687214772632298076650686, −9.90580057434869306643780012378, −9.00860704004180352853768393778, −7.43735472240707665064445190638, −5.66031191061953818894452716694, −5.05921285492872003948959002657, −3.83733799095364427444160031493, −2.82251058372178622060232979914, −1.004686489817905578785786339, 1.067220873296096900229185364183, 3.08375193628645750975427342714, 4.60705861681860144904920198261, 5.72617173236273251867044504676, 6.43114288271161294649092521572, 7.5396715976201074282921087299, 8.34575231897884175189697574511, 9.60773781589333679772108052196, 11.13207552281016468041575732604, 12.064069063577661865977117782551, 13.184983586640779247121779210297, 13.826104407529749146576271270301, 14.85216199864514360401443676725, 16.341553423528724798465998790691, 16.59343309426994131102915443101, 17.731492329753979712848051772881, 18.6866535909180651557871636702, 19.18713767435420793309940300424, 21.01687606141251519363861003657, 21.90374955025620853257192509685, 22.86266450378996119841239084849, 23.601950769552549218517461921714, 24.35308549855040938374317383762, 25.00127948565210869384049666627, 26.14378134012921237805959958225

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