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Properties

Label 1-532-532.139-r0-0-0
Degree $1$
Conductor $532$
Sign $0.775 + 0.631i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + (0.766 + 0.642i)3-s + (0.939 − 0.342i)5-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + 37-s − 39-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (0.939 − 0.342i)5-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + 37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.003391893 + 0.7120325641i\)
\(L(\frac12)\) \(\approx\) \(2.003391893 + 0.7120325641i\)
\(L(1)\) \(\approx\) \(1.540948447 + 0.3180977607i\)
\(L(1)\) \(\approx\) \(1.540948447 + 0.3180977607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−23.31367762206232036850952522526, −22.580080893218635542644276089672, −21.66287940506813276452892728468, −20.676797877961793001039541194220, −20.091322517842899758976734423715, −19.17927571904033304516114655543, −18.236640807969800751973118228776, −17.67582727240215652255782569547, −16.86690838873275821101273337435, −15.402474574316007131075087108668, −14.67831916979426060773689453502, −14.03889052445702983533935968708, −13.0932018107345813364025221738, −12.46806168762854214983053585889, −11.35063978581717092349691997414, −9.97324636062006080125773603477, −9.52274146665743588170474454580, −8.50617953149943247834448836307, −7.261603935019594393374479209807, −6.83037762589360514396010972573, −5.62011249187221937916365820911, −4.45709070213802823211351357869, −2.96180679032113534686160553217, −2.36206467206187765833882985926, −1.18905795231232842410981883390, 1.49339419893478805416766128299, 2.50008849824490323519128161533, 3.61374397437717295330611593838, 4.653948998353593255099363663729, 5.59980610629567782818062259549, 6.67587371577447553061828579629, 7.93313623271050955617850969491, 9.00700749724276529588570281933, 9.34619667346312241855159529101, 10.42019870178855516604319521843, 11.23794153839263208329607637151, 12.62933187838838111082716620840, 13.41016944211672067358508154806, 14.29028135885706260821519289908, 14.790673501183697329776541139311, 15.99349733295855210094651971244, 16.81573085377703745567917979807, 17.35749904891376069515619036640, 18.76743649900368660100787606756, 19.39709511699961279924664610434, 20.31363925297744792469819097265, 21.12547920486581775395175680862, 21.86594150693017932205497266061, 22.1916062693878659483380618270, 23.86485150974766047986907640615

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