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Properties

Label 1-245-245.124-r1-0-0
Degree $1$
Conductor $245$
Sign $0.958 - 0.284i$
Analytic cond. $26.3289$
Root an. cond. $26.3289$
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

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

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(26.3289\)
Root analytic conductor: \(26.3289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (1:\ ),\ 0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.262362546 - 0.7644416652i\)
\(L(\frac12)\) \(\approx\) \(5.262362546 - 0.7644416652i\)
\(L(1)\) \(\approx\) \(2.728791948 - 0.2248673418i\)
\(L(1)\) \(\approx\) \(2.728791948 - 0.2248673418i\)

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

−25.96163253568859989352246172278, −24.75418957679462339530477666618, −24.23960077977694660926394046570, −23.069255691561336415592024908111, −21.96628626037413126853676506038, −21.40865166258398711508805072489, −20.64373507461767188328931495131, −19.48146285393637029598131838001, −19.11967308464420712304932752156, −17.26960476256669467823841925996, −16.009482064968222150737612265001, −15.57771690348279662569879449172, −14.36873446730252844960437263196, −13.75126061649368079997367217651, −12.944654305299683427592590831615, −11.42302261334172074745310138471, −10.84805527683750821773721921479, −9.46889875693301321959980865728, −8.50065855190651929312255657304, −7.15333342473549650513924930010, −6.00742281840201406158361198034, −4.71362857153552800630102725448, −3.78501489901738950804740785855, −2.85752434352500112499066176641, −1.523315888943466384695487943297, 1.36887048527709452378295530821, 2.59254720433793197018928268385, 3.607320997040080635293067344888, 4.77665141288928091315905367278, 6.18378343940428580225265453376, 7.12500351693682837475660555126, 7.99410402748676882488754787169, 9.23882565615733772753169595461, 10.601964243130189051456018339370, 11.907232824962346539143420256875, 12.7253427016549505179186435956, 13.501565858747627773750281801712, 14.45986906601215981521274093498, 15.17691753211945195235294496940, 16.10853168901866049463300791881, 17.47076530235015167563547054225, 18.45634091885245270776684659608, 19.68599623960442788933301820642, 20.48634615702254043580663936305, 20.94331557526353384715936968622, 22.516156854401181154178292897176, 22.9277433109272488217307271617, 24.17579005998625714996482974716, 24.84919604428860112374693186451, 25.52038849526122149952602984668

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