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Properties

Label 1-245-245.103-r0-0-0
Degree $1$
Conductor $245$
Sign $-0.272 - 0.962i$
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  + (−0.294 − 0.955i)2-s + (0.930 − 0.365i)3-s + (−0.826 + 0.563i)4-s + (−0.623 − 0.781i)6-s + (0.781 + 0.623i)8-s + (0.733 − 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.563 + 0.826i)12-s + (0.974 + 0.222i)13-s + (0.365 − 0.930i)16-s + (0.997 + 0.0747i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s + (−0.433 + 0.900i)22-s + (−0.997 + 0.0747i)23-s + (0.955 + 0.294i)24-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)2-s + (0.930 − 0.365i)3-s + (−0.826 + 0.563i)4-s + (−0.623 − 0.781i)6-s + (0.781 + 0.623i)8-s + (0.733 − 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.563 + 0.826i)12-s + (0.974 + 0.222i)13-s + (0.365 − 0.930i)16-s + (0.997 + 0.0747i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s + (−0.433 + 0.900i)22-s + (−0.997 + 0.0747i)23-s + (0.955 + 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ -0.272 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7962119112 - 1.052763809i\)
\(L(\frac12)\) \(\approx\) \(0.7962119112 - 1.052763809i\)
\(L(1)\) \(\approx\) \(0.9472225377 - 0.6706174823i\)
\(L(1)\) \(\approx\) \(0.9472225377 - 0.6706174823i\)

Euler product

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

−26.06107731301868321674611226918, −25.58705121749113619530903332442, −24.91970251186545260049686338165, −23.650135363172075170188099411345, −23.04085505573289334568002387066, −21.7489126106102678660301858104, −20.777359955969803000762814414601, −19.86362258818980388154518050296, −18.7011377065953290512334864767, −18.16610677067337985578424047317, −16.78769917649547370348961624764, −15.89986269021505036956942146928, −15.22714152969077511289322803127, −14.25436704137617134729978564622, −13.51952586755261247783175541395, −12.40202019190358893436717428883, −10.35147229975022337145951985669, −10.01132575108656102989218894438, −8.56768494235585011687985435343, −8.08977630146727060504243826510, −6.974429540793934218447585838533, −5.64390118583919699094406069552, −4.509846054451874076119115172244, −3.36557751819750041039253537884, −1.65182429584638889818623682916, 1.072494885302114672592196710220, 2.41771767015261632914163413551, 3.32876565468864506219120542682, 4.422983960430282600526627226935, 6.12588489315471075440698132400, 7.777698563007684174832442332268, 8.36021707890532354369524324276, 9.39273844280267469862262205842, 10.37816035648556676558647593260, 11.465905349913661045835463715220, 12.57489079896535512275993347708, 13.48123478066517801759162803759, 14.04961474803856542509421910582, 15.42127597581338895944861746148, 16.56209983729837477226143396376, 17.93735269378081369023423923220, 18.60159585093339525808281894608, 19.36096922587330900174393631895, 20.2323672031199194643991631207, 21.17026993984259837041118724028, 21.62201070047422555495355493651, 23.18227039080342040733732362785, 23.86097600383354675553966905476, 25.20266810975135993511186903424, 26.10384856890282514738675457748

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