Properties

Label 1-35e2-1225.23-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

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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: $\chi_{1225} (23, \cdot )$
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 \( 1 + (-0.587 + 0.809i)T \)
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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

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

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