↑

Properties

Label 1-35e2-1225.1012-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.524 + 0.851i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (0.894 − 0.447i)2-s + (0.0299 + 0.999i)3-s + (0.599 − 0.800i)4-s + (0.473 + 0.880i)6-s + (0.178 − 0.983i)8-s + (−0.998 + 0.0598i)9-s + (0.998 + 0.0598i)11-s + (0.817 + 0.575i)12-s + (−0.351 − 0.936i)13-s + (−0.280 − 0.959i)16-s + (0.119 + 0.992i)17-s + (−0.866 + 0.5i)18-s + (−0.913 + 0.406i)19-s + (0.919 − 0.393i)22-s + (−0.907 − 0.420i)23-s + (0.988 + 0.149i)24-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)2-s + (0.0299 + 0.999i)3-s + (0.599 − 0.800i)4-s + (0.473 + 0.880i)6-s + (0.178 − 0.983i)8-s + (−0.998 + 0.0598i)9-s + (0.998 + 0.0598i)11-s + (0.817 + 0.575i)12-s + (−0.351 − 0.936i)13-s + (−0.280 − 0.959i)16-s + (0.119 + 0.992i)17-s + (−0.866 + 0.5i)18-s + (−0.913 + 0.406i)19-s + (0.919 − 0.393i)22-s + (−0.907 − 0.420i)23-s + (0.988 + 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.524 + 0.851i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1012, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8638080302 + 1.546119707i\)
\(L(\frac12)\) \(\approx\) \(0.8638080302 + 1.546119707i\)
\(L(1)\) \(\approx\) \(1.520074255 + 0.1133798389i\)
\(L(1)\) \(\approx\) \(1.520074255 + 0.1133798389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.894 - 0.447i)T \)
3 \( 1 + (0.0299 + 0.999i)T \)
11 \( 1 + (0.998 + 0.0598i)T \)
13 \( 1 + (-0.351 - 0.936i)T \)
17 \( 1 + (0.119 + 0.992i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.907 - 0.420i)T \)
29 \( 1 + (-0.753 - 0.657i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.817 + 0.575i)T \)
41 \( 1 + (0.134 + 0.990i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.701 + 0.712i)T \)
53 \( 1 + (0.800 + 0.599i)T \)
59 \( 1 + (-0.971 - 0.237i)T \)
61 \( 1 + (0.0149 + 0.999i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (-0.986 + 0.163i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.266 + 0.963i)T \)
89 \( 1 + (0.772 - 0.635i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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

−20.61294466291770612855515608576, −20.003710165920005694507356687558, −19.21089961046241124001086115440, −18.38440597537251543594087442537, −17.397394574156522371260990404030, −16.858458663347076335458132791969, −16.13841286005650228416200706062, −14.9504862442914933651397196482, −14.40173387552264893472450160484, −13.69964997585621904937226015506, −13.09241917160458423508827666948, −11.98210422681589136428919998632, −11.83343753543911080916228293892, −10.87842988609228635808805969698, −9.31058546343619294320526217537, −8.66376616085039199739122145323, −7.5067114913770771076332410010, −7.06007045521202809954236273555, −6.25324364161850226886296830519, −5.52125503729155309852051609077, −4.39221852726745326049440714874, −3.58678972845423077787288523799, −2.38397261104905670830604400060, −1.76452183206848433375169655072, −0.22697451567720373279780029423, 1.26220569106368457934773686193, 2.42941904391503892280029990315, 3.363518963267696684401641110125, 4.137435619551692405596963132056, 4.71087613932787302776715460286, 5.94620699443900795237519520682, 6.20595773350895212856921505277, 7.69297819403506766206825039845, 8.68862718976609170746502392406, 9.71154459880131347077341607400, 10.31904864433863047386040221262, 10.9792348225008601236535123805, 11.88702354013427470878139449891, 12.55416240572417236387987879262, 13.47359738772383159114123050024, 14.54390613586992110635546904699, 14.80300151563700632173498095161, 15.532679204165218189181223735023, 16.53638517886089548388368468740, 17.065047411166363724567628313501, 18.175612588011358210575746945236, 19.4388595450424641599996133391, 19.79472766895696005937241144694, 20.54538375511685654551095243613, 21.348559320952654140049287296703

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