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Properties

Label 2-525-105.59-c1-0-27
Degree $2$
Conductor $525$
Sign $0.978 - 0.208i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.192 + 0.334i)2-s + (0.983 + 1.42i)3-s + (0.925 − 1.60i)4-s + (−0.286 + 0.603i)6-s + (1.17 − 2.36i)7-s + 1.48·8-s + (−1.06 + 2.80i)9-s + (−2.20 − 1.27i)11-s + (3.19 − 0.257i)12-s + 3.06·13-s + (1.01 − 0.0640i)14-s + (−1.56 − 2.71i)16-s + (5.59 + 3.23i)17-s + (−1.14 + 0.185i)18-s + (1.03 − 0.597i)19-s + ⋯
L(s)  = 1  + (0.136 + 0.236i)2-s + (0.567 + 0.823i)3-s + (0.462 − 0.801i)4-s + (−0.116 + 0.246i)6-s + (0.444 − 0.895i)7-s + 0.525·8-s + (−0.354 + 0.934i)9-s + (−0.663 − 0.383i)11-s + (0.922 − 0.0743i)12-s + 0.850·13-s + (0.272 − 0.0171i)14-s + (−0.391 − 0.677i)16-s + (1.35 + 0.783i)17-s + (−0.269 + 0.0436i)18-s + (0.237 − 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.978 - 0.208i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.978 - 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15760 + 0.227338i\)
\(L(\frac12)\) \(\approx\) \(2.15760 + 0.227338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.983 - 1.42i)T \)
5 \( 1 \)
7 \( 1 + (-1.17 + 2.36i)T \)
good2 \( 1 + (-0.192 - 0.334i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.20 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + (-5.59 - 3.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 + 0.597i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.52 + 2.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.77iT - 29T^{2} \)
31 \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.07 + 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 5.46iT - 43T^{2} \)
47 \( 1 + (2.78 - 1.61i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.62 - 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.08 - 4.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.04 + 1.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.921iT - 71T^{2} \)
73 \( 1 + (0.148 - 0.256i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.14 - 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.11iT - 83T^{2} \)
89 \( 1 + (9.41 + 16.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74590295087019853039054738067, −10.21570396165560305507474871600, −9.234981518114710468818618179507, −8.078678634804023433802971732627, −7.47667844030045916911906800810, −6.08301983181701254678478100674, −5.26850368506769282473756006847, −4.22079322376156782669131624098, −3.10426223594588739677816053834, −1.46039126370814782193244410126, 1.71491983708777816774014428603, 2.75012672877664786032607930903, 3.67097796639943696171993881873, 5.29318175759614676944341691977, 6.33828160096147305859439495228, 7.58135515583743138981588455839, 7.907664400808571850283326962123, 8.840101871864182211855591731704, 9.852531691786581298092086798508, 11.19968060776270229780070030820

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