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Properties

Label 2-525-15.8-c1-0-2
Degree $2$
Conductor $525$
Sign $0.998 - 0.0457i$
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  + (−1.24 − 1.24i)2-s + (−0.474 − 1.66i)3-s + 1.09i·4-s + (−1.48 + 2.66i)6-s + (−0.707 + 0.707i)7-s + (−1.12 + 1.12i)8-s + (−2.54 + 1.58i)9-s + 1.55i·11-s + (1.82 − 0.520i)12-s + (4.50 + 4.50i)13-s + 1.75·14-s + 4.99·16-s + (−2.13 − 2.13i)17-s + (5.13 + 1.20i)18-s + 4.20i·19-s + ⋯
L(s)  = 1  + (−0.879 − 0.879i)2-s + (−0.274 − 0.961i)3-s + 0.547i·4-s + (−0.604 + 1.08i)6-s + (−0.267 + 0.267i)7-s + (−0.397 + 0.397i)8-s + (−0.849 + 0.527i)9-s + 0.468i·11-s + (0.526 − 0.150i)12-s + (1.25 + 1.25i)13-s + 0.470·14-s + 1.24·16-s + (−0.517 − 0.517i)17-s + (1.21 + 0.283i)18-s + 0.965i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.998 - 0.0457i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.998 - 0.0457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487523 + 0.0111537i\)
\(L(\frac12)\) \(\approx\) \(0.487523 + 0.0111537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.474 + 1.66i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.24 + 1.24i)T + 2iT^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 + (-4.50 - 4.50i)T + 13iT^{2} \)
17 \( 1 + (2.13 + 2.13i)T + 17iT^{2} \)
19 \( 1 - 4.20iT - 19T^{2} \)
23 \( 1 + (3.76 - 3.76i)T - 23iT^{2} \)
29 \( 1 + 2.97T + 29T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 + (-1.23 + 1.23i)T - 37iT^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (-2.09 - 2.09i)T + 43iT^{2} \)
47 \( 1 + (0.0358 + 0.0358i)T + 47iT^{2} \)
53 \( 1 + (-4.30 + 4.30i)T - 53iT^{2} \)
59 \( 1 - 4.93T + 59T^{2} \)
61 \( 1 - 3.31T + 61T^{2} \)
67 \( 1 + (1.71 - 1.71i)T - 67iT^{2} \)
71 \( 1 - 5.73iT - 71T^{2} \)
73 \( 1 + (7.26 + 7.26i)T + 73iT^{2} \)
79 \( 1 - 3.59iT - 79T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 83iT^{2} \)
89 \( 1 - 1.35T + 89T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−11.11597287199695651434669817645, −9.959530606753676432615165088519, −9.135468376918429966060403943719, −8.426083229589887033023184889080, −7.40101483821293849825141085488, −6.34012697397587763065158012291, −5.52984331035660836795864538617, −3.76822474903103671917030077281, −2.26472470141127432062953262300, −1.43539914615005691274686843167, 0.41265831363990977124167482396, 3.17035319762234621974803232153, 4.10423440071694293826129021331, 5.64823400207173618229312252501, 6.19168192771035859344528345464, 7.31676431693545759420076497534, 8.521613077851563526210872439862, 8.781272345398036643074588047799, 9.895562173615250149541149469376, 10.61792872206581045110676328116

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