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Properties

Label 2-525-15.8-c1-0-17
Degree $2$
Conductor $525$
Sign $0.175 - 0.984i$
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  + (1.11 + 1.11i)2-s + (1.51 + 0.843i)3-s + 0.487i·4-s + (0.746 + 2.62i)6-s + (−0.707 + 0.707i)7-s + (1.68 − 1.68i)8-s + (1.57 + 2.55i)9-s + 2.87i·11-s + (−0.411 + 0.737i)12-s + (1.36 + 1.36i)13-s − 1.57·14-s + 4.73·16-s + (−4.87 − 4.87i)17-s + (−1.08 + 4.60i)18-s + 3.03i·19-s + ⋯
L(s)  = 1  + (0.788 + 0.788i)2-s + (0.873 + 0.486i)3-s + 0.243i·4-s + (0.304 + 1.07i)6-s + (−0.267 + 0.267i)7-s + (0.596 − 0.596i)8-s + (0.525 + 0.850i)9-s + 0.865i·11-s + (−0.118 + 0.212i)12-s + (0.378 + 0.378i)13-s − 0.421·14-s + 1.18·16-s + (−1.18 − 1.18i)17-s + (−0.256 + 1.08i)18-s + 0.697i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.175 - 0.984i$
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: \((2,\ 525,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18313 + 1.82895i\)
\(L(\frac12)\) \(\approx\) \(2.18313 + 1.82895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 - 0.843i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.11 - 1.11i)T + 2iT^{2} \)
11 \( 1 - 2.87iT - 11T^{2} \)
13 \( 1 + (-1.36 - 1.36i)T + 13iT^{2} \)
17 \( 1 + (4.87 + 4.87i)T + 17iT^{2} \)
19 \( 1 - 3.03iT - 19T^{2} \)
23 \( 1 + (-4.40 + 4.40i)T - 23iT^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 + (4.41 - 4.41i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (4.59 + 4.59i)T + 43iT^{2} \)
47 \( 1 + (6.91 + 6.91i)T + 47iT^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 + 2.66T + 59T^{2} \)
61 \( 1 - 2.03T + 61T^{2} \)
67 \( 1 + (-11.0 + 11.0i)T - 67iT^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (4.85 + 4.85i)T + 73iT^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 + (7.16 - 7.16i)T - 83iT^{2} \)
89 \( 1 - 0.776T + 89T^{2} \)
97 \( 1 + (1.56 - 1.56i)T - 97iT^{2} \)
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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

−10.88878855160787244100454159833, −10.02289504419459374194362044670, −9.241609524884490955638535869334, −8.361570078404043166834464639438, −7.14646839191945518589124035323, −6.64386773220444029618201119352, −5.20157320729372020511962794194, −4.56037413785837609451473093512, −3.51988252807989890557112781021, −2.08828984464108470204108551375, 1.51092874851301032781498242978, 2.86889484881180433877899964036, 3.54954112587488433886831356028, 4.57506894331881874170337143762, 6.00120825857639566833437668587, 7.04579411643644784296973352319, 8.142711977074050826416114048380, 8.735857959601727437432538231845, 9.877268543220167844839191148741, 11.04478245282170296988769726765

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