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Properties

Label 1-525-525.2-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: $\chi_{525} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8214833328 - 0.2291217125i\)
\(L(\frac12)\) \(\approx\) \(0.8214833328 - 0.2291217125i\)
\(L(1)\) \(\approx\) \(0.8634126599 + 0.2346957950i\)
\(L(1)\) \(\approx\) \(0.8634126599 + 0.2346957950i\)

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

−23.3271506624380797603910317730, −22.804622736748270164529710358539, −21.61544521293595102561806417642, −21.18404317530268345279636713967, −20.16140797262154679817713229446, −19.64299016819236451984716868484, −18.41347714576433407861967269037, −18.08260248331464883278010828028, −16.99527275887692884788014031839, −15.76635844117541627243182538796, −14.9433544365995407443639452304, −13.9334763110319110668063541396, −13.087249613089820310059499494484, −12.49470185653969415151481530745, −11.29486037879534643822376580035, −10.775928484998700764188908590866, −9.72916533939162093502617718183, −8.90308760941863308040515069162, −7.930485838789546681831315502051, −6.57845533991510485505457234086, −5.471253221715539957719387461514, −4.46908822655562290147901139598, −3.60345775005106115738306592454, −2.37373101998124321803690429307, −1.471503467053628274545089206082, 0.42864351330238162209149811673, 2.42732095079595989312077379565, 3.72916208455423310876195707562, 4.59121940922351136595954660226, 5.80333665212231099618247146152, 6.36087269604734549572424616567, 7.53559897701008883020957046488, 8.45138599958532763724931540229, 9.02782407934600719996943332888, 10.39377472301498952486141477329, 11.23047899908180417106096652001, 12.59797130965202372962346001581, 13.341921468672433131816225922738, 13.9306332619684230624518841921, 15.258712340779256271926229943, 15.53995182201806701788098402950, 16.66037205842351334928689572160, 17.268889604299892903792491164343, 18.46677966136861312571212001864, 18.740475747062119373215219940755, 20.192552749018447727433831958333, 21.09225363472946796787537995042, 21.97577537361568853801600902454, 22.6802291116208039170268747929, 23.67006616021966584049598035760

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