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Properties

Label 1-525-525.23-r0-0-0
Degree $1$
Conductor $525$
Sign $-0.855 + 0.517i$
Analytic cond. $2.43808$
Root an. cond. $2.43808$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)11-s + (−0.951 + 0.309i)13-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.587 + 0.809i)22-s + (0.207 + 0.978i)23-s + (0.5 + 0.866i)26-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.866 − 0.5i)32-s + (−0.309 − 0.951i)34-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)11-s + (−0.951 + 0.309i)13-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.587 + 0.809i)22-s + (0.207 + 0.978i)23-s + (0.5 + 0.866i)26-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.866 − 0.5i)32-s + (−0.309 − 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(2.43808\)
Root analytic conductor: \(2.43808\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ -0.855 + 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08213797978 - 0.2944940513i\)
\(L(\frac12)\) \(\approx\) \(-0.08213797978 - 0.2944940513i\)
\(L(1)\) \(\approx\) \(0.5531039433 - 0.3386393905i\)
\(L(1)\) \(\approx\) \(0.5531039433 - 0.3386393905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.207 + 0.978i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (-0.406 - 0.913i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
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

−24.00992606587413391552225550213, −23.272977176525350098638380546512, −22.60176932196320180112695252630, −21.654638096000914605597644134133, −20.64297946170289895028984596368, −19.6101904995882612383880476597, −18.708483576382856057902440968100, −18.04245124681263571721675185975, −17.017189605042420667494356214888, −16.55551725081748047652319760127, −15.34529349837102974486645499991, −14.83254538754404032162837721987, −14.00870439733437603750865708688, −12.79204247080523480938234192288, −12.31996047642968527772577546301, −10.56307899610008294736407129656, −10.06241248987915668226866532094, −8.98585040013490594581724549097, −7.99762960915695094611532415773, −7.30307611982766700753300986138, −6.331117429263286850132382835507, −5.21615200947858717908164650166, −4.57609361965088226359916714860, −3.16826086322360833892128014132, −1.65183281885379558835955226035, 0.17077769358703217033428417077, 1.74425027494239855017719259435, 2.774080892021285273922196419559, 3.73446948108194186857633053132, 4.87914936402102108839905928145, 5.77677517787545520407665734762, 7.368955307203984853540791866306, 8.139272805829422819643849325167, 9.226702502520862223909853798226, 9.95309003440875338026142220175, 10.92375462979926732169567799933, 11.65040983217398482373145109751, 12.61243246415025514195714803978, 13.364602371391852902060696863877, 14.249200605327699184769322437543, 15.22471312791585943152581693080, 16.53024736334018666920858901215, 17.17318092019767419590654012732, 18.11535090771764534317661226272, 19.17418425478715155288755107228, 19.33445991139151159863806107853, 20.65144816986232655216096174607, 21.258090922204246174440564925660, 21.92206635177229635936109781087, 22.89086765304522587840037889457

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