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Properties

Label 1-245-245.89-r1-0-0
Degree $1$
Conductor $245$
Sign $0.843 - 0.536i$
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  + (−0.0747 − 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (0.222 − 0.974i)6-s + (0.222 + 0.974i)8-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.623 − 0.781i)22-s + (−0.365 + 0.930i)23-s + (−0.0747 + 0.997i)24-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (0.222 − 0.974i)6-s + (0.222 + 0.974i)8-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.623 − 0.781i)22-s + (−0.365 + 0.930i)23-s + (−0.0747 + 0.997i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(26.3289\)
Root analytic conductor: \(26.3289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (1:\ ),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.379538168 - 0.6924228092i\)
\(L(\frac12)\) \(\approx\) \(2.379538168 - 0.6924228092i\)
\(L(1)\) \(\approx\) \(1.358001226 - 0.4199335104i\)
\(L(1)\) \(\approx\) \(1.358001226 - 0.4199335104i\)

Euler product

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

−25.85384789599107722319108825260, −24.94110413591273918579983489196, −24.5004934381432103124830161154, −23.47584869348376172920142222705, −22.406858521099726979866405995433, −21.513445643262606902676183307167, −20.13540463844253488769097005203, −19.45249738156520610065527386752, −18.38472214084891961686983402729, −17.61815597735087578131269599787, −16.46111656703449116993999876872, −15.541664998621765157139045980834, −14.41247683913594219798695338146, −14.167884717748722160946916526535, −12.89322419792339461533410733775, −11.96484448447050487804693167852, −9.99208928400588686671673654900, −9.29565147192942355894589433836, −8.39418427598559164423932336180, −7.17270524247875448944531279031, −6.76218182770630634994137379855, −5.03748517850814585392116532127, −4.06116941533897109469191198822, −2.59858930583666627778675543908, −0.92336780286390238657474310678, 1.1420091981635245284899762869, 2.40506966581284979821745418171, 3.50186037439036535520379300636, 4.31731432698079159002462813564, 5.76586592380433330299615392139, 7.682316132582796596144122054432, 8.42587450776774795688128386503, 9.62526165171236768835771798801, 10.10692877715623662325510209125, 11.443042477496552852192762967317, 12.4273943663968660310830004403, 13.473899986426640441784915851037, 14.30040067484334046003973162554, 15.10464375945272187552474888936, 16.561652214678080286417655087184, 17.53563158775531019434875060972, 18.81256276973917402089145947774, 19.47332831831213830405011270077, 20.12527425750767512432095985021, 21.16086794598849750975241185508, 21.82610185202668863477768918928, 22.668200838942903896871910333098, 24.01501242444942396835604420975, 25.01161346477850523333613849533, 25.99706753437726732191697237969

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