↑

Properties

Label 1-245-245.243-r0-0-0
Degree $1$
Conductor $245$
Sign $-0.488 - 0.872i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5078057461 - 0.8667204905i\)
\(L(\frac12)\) \(\approx\) \(0.5078057461 - 0.8667204905i\)
\(L(1)\) \(\approx\) \(0.8663152627 - 0.4910122992i\)
\(L(1)\) \(\approx\) \(0.8663152627 - 0.4910122992i\)

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 \( 1 + iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.98270809385845784159579897417, −25.31614710409830322489827827115, −24.55922071273193554294856832474, −23.483169901474874392332722063964, −23.04992852979827730721023536585, −22.18365206999272692643853359259, −21.16391042901169975012666218404, −19.933690989439485552453375062395, −18.68331272515747148645765552610, −17.65422291022943395003242045269, −17.2484053301875479385181517237, −16.0848577184411496856114764808, −15.14951964497798746802954568994, −14.1408276802011916570185514812, −12.86883029963037706359412219316, −12.62251838206897355780286554569, −11.38029155858348198369553194645, −10.13707531108267780030321458870, −8.49560705640589629354299752860, −7.64570101626901493474922427887, −6.6692935287837900013327990337, −5.771389765284123409365854110332, −4.822237787532672071949741745222, −3.45831458515962896768620626811, −1.73787974239153286119517822, 0.65872606585899260169724590476, 2.5188418351300273919501511213, 3.81950459620709118702849089460, 4.65223901791442300247469414401, 5.77014851463653407552599508786, 6.66820450784506286320023348607, 8.81045472802040439274056814091, 9.52442427612754303287510145835, 10.81543375420990066878048972986, 11.286322391071413422369171487809, 12.211206934457612860013223822620, 13.456073177635336054877964546117, 14.30932722419083105313670576709, 15.46167254981188432731667990450, 16.28856669682656964038552991194, 17.419334035347219254885613993122, 18.576593797264215045868571368818, 19.33449527855566062261758759211, 20.790091624165640624404560998817, 21.05520180484674288239640311510, 22.22208036629956073689987241060, 22.66607714413518504979493376993, 23.83610728350100398224125367800, 24.39788177535783194515984374443, 26.16188158494365234455743178270

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