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Properties

Label 1-532-532.59-r1-0-0
Degree $1$
Conductor $532$
Sign $-0.809 - 0.586i$
Analytic cond. $57.1713$
Root an. cond. $57.1713$
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 + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s − 31-s + (0.173 + 0.984i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s − 31-s + (0.173 + 0.984i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ -0.809 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3182563678 - 0.9813899787i\)
\(L(\frac12)\) \(\approx\) \(0.3182563678 - 0.9813899787i\)
\(L(1)\) \(\approx\) \(0.7789847190 - 0.2269930035i\)
\(L(1)\) \(\approx\) \(0.7789847190 - 0.2269930035i\)

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 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−23.537625948825384829088283467708, −22.82917618103625400689873160077, −22.05814409115419679295565826048, −21.39755412500571407109918467950, −19.98363389565933916467493764697, −19.16736256701713250174738405541, −18.60012130677208560067972009690, −17.644067934131104773772451394212, −17.0883986412665730958429832945, −16.02268210285431096482650184631, −14.993351482396988135482892752, −14.23172401483666456379700473838, −13.22536075093547639551560364086, −12.28449669167711243860729955644, −11.50002486535867517907455234260, −10.784588200526490081233427129488, −9.876192551278776260087304602756, −8.60765645198468695088004021623, −7.34934250511576054754511921435, −6.820298836622097060301480178623, −6.05577660039478779685804906675, −4.78583403634886930651313595256, −3.7328268181105201044088699502, −2.29952769696700753048763165944, −1.37893537519913187232447199762, 0.34603689966718457679383838681, 1.07514460467544830876405931282, 3.02040654618698508866853416367, 4.07841505564354075640738470858, 5.04086013406815810671975415752, 5.702557963237108932553510439645, 6.7881521354285322294990009356, 8.13344900417067596014502462323, 9.01545957724611780680750492483, 9.78012469925966987206153677887, 10.94659502300499298110707260396, 11.56114515015733980452035754607, 12.52430921769811111383184195977, 13.28984941069588110633548319546, 14.50444671422070930744162606939, 15.55023671132508009125129683160, 16.21999730710525146354079507698, 16.84287069806783240541651532639, 17.66987099649680563240430601480, 18.56809916854874258728069670387, 19.833350082087758398252832869110, 20.44251998176800395646781427371, 21.34155262602209008651870656369, 21.99762715862884197096269723847, 23.059387565688801937270213979055

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