Properties

Label 2-525-35.27-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.152 - 0.988i$
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  + (−0.653 − 0.653i)2-s + (−0.707 − 0.707i)3-s − 1.14i·4-s + 0.924i·6-s + (0.741 + 2.53i)7-s + (−2.05 + 2.05i)8-s + 1.00i·9-s − 0.746·11-s + (−0.809 + 0.809i)12-s + (−4.26 − 4.26i)13-s + (1.17 − 2.14i)14-s + 0.398·16-s + (−5.29 + 5.29i)17-s + (0.653 − 0.653i)18-s − 1.17·19-s + ⋯
L(s)  = 1  + (−0.462 − 0.462i)2-s + (−0.408 − 0.408i)3-s − 0.572i·4-s + 0.377i·6-s + (0.280 + 0.959i)7-s + (−0.726 + 0.726i)8-s + 0.333i·9-s − 0.225·11-s + (−0.233 + 0.233i)12-s + (−1.18 − 1.18i)13-s + (0.314 − 0.573i)14-s + 0.0996·16-s + (−1.28 + 1.28i)17-s + (0.154 − 0.154i)18-s − 0.269·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.152 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100980 + 0.117721i\)
\(L(\frac12)\) \(\approx\) \(0.100980 + 0.117721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-0.741 - 2.53i)T \)
good2 \( 1 + (0.653 + 0.653i)T + 2iT^{2} \)
11 \( 1 + 0.746T + 11T^{2} \)
13 \( 1 + (4.26 + 4.26i)T + 13iT^{2} \)
17 \( 1 + (5.29 - 5.29i)T - 17iT^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + (3.19 - 3.19i)T - 23iT^{2} \)
29 \( 1 + 2.45iT - 29T^{2} \)
31 \( 1 - 4.87iT - 31T^{2} \)
37 \( 1 + (-2.05 - 2.05i)T + 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (-6.64 + 6.64i)T - 43iT^{2} \)
47 \( 1 + (-5.29 + 5.29i)T - 47iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 53iT^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 2.52iT - 61T^{2} \)
67 \( 1 + (6.47 + 6.47i)T + 67iT^{2} \)
71 \( 1 + 6.74T + 71T^{2} \)
73 \( 1 + (2.47 + 2.47i)T + 73iT^{2} \)
79 \( 1 - 5.83iT - 79T^{2} \)
83 \( 1 + (-0.768 - 0.768i)T + 83iT^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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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

−10.91538118666965092427007805885, −10.40853242113916597131684120991, −9.418509250106809317325821922125, −8.528700101749762151503816459030, −7.69979390887276017422219964982, −6.29401373709057344120163630231, −5.64869003322377715884182544054, −4.70668071862849210438411243346, −2.71231817539502824723049713258, −1.77208999439373727204221284685, 0.10419376464684427957214884108, 2.53407248013544587224358591277, 4.16331943965716645924973429754, 4.66822745725875087877357313725, 6.30061416299411384205077773746, 7.13775203402436335739332947309, 7.70451189315506870399127916568, 9.007113364464341454037788968656, 9.508879086233057664115555446483, 10.58345035573634144895374359329

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