Properties

Label 1-245-245.109-r0-0-0
Degree $1$
Conductor $245$
Sign $-0.536 + 0.843i$
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  + (−0.0747 + 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

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

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.536 + 0.843i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ -0.536 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3652077246 + 0.6649835015i\)
\(L(\frac12)\) \(\approx\) \(0.3652077246 + 0.6649835015i\)
\(L(1)\) \(\approx\) \(0.5947550499 + 0.4501834847i\)
\(L(1)\) \(\approx\) \(0.5947550499 + 0.4501834847i\)

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

−25.980534592307300443030682662338, −24.7154924069226074261196818130, −23.60339068751960546874557504021, −23.00248434173201529668931532831, −21.88767116626226187263271168249, −21.49459276904254713631998189023, −20.161207207328235872421004906547, −19.2418948195574083012685416812, −18.41076765733281676659751018709, −17.59206335920300005640625133556, −16.7341086138723092988016042909, −15.632645161072971303213826858902, −13.918873269870409239857651251426, −13.37346843704234440779647202759, −12.11936716298035976576865497032, −11.44488411276158430979105186336, −10.760863030102432038817232779441, −9.51038697589109381933139445596, −8.54826353066864653902100203038, −7.04478108567624642627746755974, −5.88991700247365177652610545876, −4.70991909348787981903043764436, −3.64563443113218506041415034324, −2.0280245468408838382435777513, −0.76408089459713468471800603562, 1.27018058185987635780813143072, 3.86275912110223505722568265281, 4.603293474939971096519887809774, 6.04984681251232186324460107839, 6.38631968230665352484008759416, 7.778062109206048100221340906805, 8.91723928644787519944150884448, 10.017888035894091358061979279774, 10.918096914249243265732206196930, 12.32863437029850902376729622519, 13.07359055221195377091990107232, 14.52927428212377761751850660628, 15.21507691271692076029398803612, 16.32492594976923741298722433387, 16.886782246149585562604009684038, 17.884954335400407614806766263698, 18.49458250865540567544935804322, 19.84769605126762705959923809981, 21.21415347831093449425194182621, 22.180127367702101278544340475629, 22.88625162619724244305562573419, 23.593811620541753287761042401, 24.53960626337805155144579367057, 25.47319449685552311080910455415, 26.38228133197030610935647533390

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