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Properties

Label 1-245-245.184-r0-0-0
Degree $1$
Conductor $245$
Sign $0.284 - 0.958i$
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  + (−0.365 + 0.930i)2-s + (−0.0747 + 0.997i)3-s + (−0.733 − 0.680i)4-s + (−0.900 − 0.433i)6-s + (0.900 − 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.988 + 0.149i)11-s + (0.733 − 0.680i)12-s + (−0.623 − 0.781i)13-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.222 − 0.974i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)2-s + (−0.0747 + 0.997i)3-s + (−0.733 − 0.680i)4-s + (−0.900 − 0.433i)6-s + (0.900 − 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.988 + 0.149i)11-s + (0.733 − 0.680i)12-s + (−0.623 − 0.781i)13-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.222 − 0.974i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.284 - 0.958i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05061470548 - 0.03777473644i\)
\(L(\frac12)\) \(\approx\) \(0.05061470548 - 0.03777473644i\)
\(L(1)\) \(\approx\) \(0.4353521850 + 0.2975804811i\)
\(L(1)\) \(\approx\) \(0.4353521850 + 0.2975804811i\)

Euler product

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

−26.25145003861251311776700231098, −25.65417496648232605033199818661, −24.264432428542809361121727034721, −23.61517041885268574255325930154, −22.488466381397452664510383220930, −21.663860134387632491012476331233, −20.506408178599860153916857974925, −19.763892289733928190611949425442, −18.75197773286201503273467715861, −18.28862435321599468665336343628, −17.23703278970071052042578849457, −16.38995976224280283956036503846, −14.66839372302000221985600042040, −13.61432795290988086291341425462, −12.851519470036392131637028871234, −12.01498606378944025022242422892, −11.086574503242417706596719488845, −10.07601220415764839502581570486, −8.75854819455500401370673464783, −7.966842832487217840961441652808, −6.8859483862118343471092584429, −5.45326869031462531673476922271, −4.05516085514674220651310904219, −2.544998876480896704184847077544, −1.78514426510273753314739881715, 0.047721277954355327679772400806, 2.58623701209161934980427666330, 4.25400316743027273496319066057, 5.09827628916259645799141029426, 6.04966317615309003990250905675, 7.4311768060950461178729692673, 8.42160561027935621620048019849, 9.45088180973209920830433527921, 10.27891668832499131806799041757, 11.16176702842880004793865094213, 12.84081674118492807161157433421, 13.89653978922880931964519534294, 15.0459328558710084090142698871, 15.56642279995164251147316921181, 16.37531693521251070704298268887, 17.51747201252326050753280854676, 18.00646727856697081150457015012, 19.502998777718388817334577583079, 20.25062082687041379868997375657, 21.543072665062360563982391922141, 22.34459195523732998531665964346, 23.210755529657605857423945585085, 24.12569301029422557722183444285, 25.1970002969209297608350991592, 26.09600409321460070368697843474

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