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Properties

Label 1-525-525.188-r1-0-0
Degree $1$
Conductor $525$
Sign $0.425 + 0.904i$
Analytic cond. $56.4190$
Root an. cond. $56.4190$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (0.587 + 0.809i)22-s + (0.951 − 0.309i)23-s + 26-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + i·32-s + (0.309 − 0.951i)34-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (0.587 + 0.809i)22-s + (0.951 − 0.309i)23-s + 26-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + i·32-s + (0.309 − 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(56.4190\)
Root analytic conductor: \(56.4190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (1:\ ),\ 0.425 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6319053968 + 0.4010193590i\)
\(L(\frac12)\) \(\approx\) \(0.6319053968 + 0.4010193590i\)
\(L(1)\) \(\approx\) \(0.6161751977 + 0.06182696565i\)
\(L(1)\) \(\approx\) \(0.6161751977 + 0.06182696565i\)

Euler product

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

−23.057807434324450815257251314498, −22.23158387104977397455217707947, −21.14404520297339885644556255207, −20.59532679617515327516686114920, −19.668463130751075221350432779446, −18.946954536709381765512668162966, −18.1176973968312003961472497215, −17.21186683065031390743372326286, −16.71824360400512313487428162409, −15.450058659942180954897158043494, −14.97953121333518648891474721393, −13.54446220584729022578015254391, −12.53783407323228038224170222338, −11.82778795386530940586727237626, −10.872887306105977755777083073054, −9.90903274067573412696402723079, −9.32287387289843381095526864769, −8.22627963762629866228975506948, −7.29378682063000842024968393801, −6.63383944551299330476883209787, −5.1511760122223840423497154102, −4.006188309701135542120243418771, −2.613519259462591390431942943802, −1.89058940787516892859475140942, −0.359932097166388406669502820, 0.73880473482927507651346324115, 2.125454078976411462130278418217, 3.12176258208985630358599388810, 4.74598717222144079078701372086, 5.79850427776609415700315377128, 6.72637483875779816012775540779, 7.63987292560332908594887211199, 8.62805673884105194923648637419, 9.22050012187674098204616441802, 10.59785934121136707298684051094, 10.83231516998617065759733026424, 12.11166019842128394012239590249, 13.0918053642095103374348259298, 14.293196416754709509010307550639, 15.14008566446633018619533150635, 15.84257029102852278092556667511, 16.93670130134296304158744195146, 17.35176718745268135205114985287, 18.39050760749737888539625550370, 19.30102687304418937703635538271, 19.68751224140487930920578668885, 20.88214614669672085668264223351, 21.58813049636886141148483042940, 22.6762374038736053017268574185, 23.83613159436623573035543390230

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