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Properties

Label 1-245-245.73-r0-0-0
Degree $1$
Conductor $245$
Sign $0.561 + 0.827i$
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  + (−0.563 − 0.826i)2-s + (0.680 + 0.733i)3-s + (−0.365 + 0.930i)4-s + (0.222 − 0.974i)6-s + (0.974 − 0.222i)8-s + (−0.0747 + 0.997i)9-s + (0.0747 + 0.997i)11-s + (−0.930 + 0.365i)12-s + (−0.433 + 0.900i)13-s + (−0.733 − 0.680i)16-s + (0.149 − 0.988i)17-s + (0.866 − 0.5i)18-s + (−0.5 + 0.866i)19-s + (0.781 − 0.623i)22-s + (−0.149 − 0.988i)23-s + (0.826 + 0.563i)24-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (0.680 + 0.733i)3-s + (−0.365 + 0.930i)4-s + (0.222 − 0.974i)6-s + (0.974 − 0.222i)8-s + (−0.0747 + 0.997i)9-s + (0.0747 + 0.997i)11-s + (−0.930 + 0.365i)12-s + (−0.433 + 0.900i)13-s + (−0.733 − 0.680i)16-s + (0.149 − 0.988i)17-s + (0.866 − 0.5i)18-s + (−0.5 + 0.866i)19-s + (0.781 − 0.623i)22-s + (−0.149 − 0.988i)23-s + (0.826 + 0.563i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8658653895 + 0.4587767121i\)
\(L(\frac12)\) \(\approx\) \(0.8658653895 + 0.4587767121i\)
\(L(1)\) \(\approx\) \(0.9134617615 + 0.1156711869i\)
\(L(1)\) \(\approx\) \(0.9134617615 + 0.1156711869i\)

Euler product

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

−25.883470408190599351863988514873, −25.11377915432292065586530832307, −24.1984520193820558951070958455, −23.752070453682320116197627427333, −22.54080236435537251841945482489, −21.30200696034388869937841283768, −19.910649385236478282692591448432, −19.3865773029182603754476621641, −18.52907402098925596292064181582, −17.54961097809921208226369827573, −16.82292365699444623080717336268, −15.35575041440776724440476579315, −14.91797548233900121654105328155, −13.64445450685303081335637876878, −13.1090248897189668706979203263, −11.57397194825327440870839550138, −10.29361148599554113718409033156, −9.181511465579555161715582953215, −8.26532632874755840956871187025, −7.587588771924600711335099038163, −6.40895051196679562437833228135, −5.546666375605639662659254947292, −3.819605510690488716188835403683, −2.2977789560338769464801200730, −0.79191730726466424431272204233, 1.81997788408183767749633082326, 2.801251683216596924744809429033, 4.089207370301987789442691351224, 4.85432571986501287801067461813, 6.99369469395133263786787104684, 8.04339647184959109200791429233, 9.11505771020936321920095263541, 9.79331543719385169129457345909, 10.6560484430086601034318055924, 11.82532883906520869508336315371, 12.77504671238080199286814150826, 14.0215145721667221917292525367, 14.802027212026674182760283116656, 16.2191524888345837308007959072, 16.804651711305623116462088896004, 18.11105796062022394247172550585, 18.97770608935231600328473737913, 19.96133762248436006851928575440, 20.580388184842656664026098741577, 21.400622473644389432769625185392, 22.26228267738325418000503237901, 23.17132774741668845446004293276, 24.88916768086029133916043063870, 25.54034302441804846043116572332, 26.53459076952504088004685261952

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