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Properties

Label 2-525-35.3-c1-0-21
Degree $2$
Conductor $525$
Sign $0.977 - 0.209i$
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  + (2.60 + 0.698i)2-s + (−0.258 − 0.965i)3-s + (4.57 + 2.64i)4-s − 2.69i·6-s + (0.543 − 2.58i)7-s + (6.26 + 6.26i)8-s + (−0.866 + 0.499i)9-s + (1.69 − 2.93i)11-s + (1.36 − 5.10i)12-s + (−4.01 + 4.01i)13-s + (3.22 − 6.36i)14-s + (6.67 + 11.5i)16-s + (−3.76 + 1.00i)17-s + (−2.60 + 0.698i)18-s + (−0.721 − 1.24i)19-s + ⋯
L(s)  = 1  + (1.84 + 0.493i)2-s + (−0.149 − 0.557i)3-s + (2.28 + 1.32i)4-s − 1.10i·6-s + (0.205 − 0.978i)7-s + (2.21 + 2.21i)8-s + (−0.288 + 0.166i)9-s + (0.511 − 0.885i)11-s + (0.394 − 1.47i)12-s + (−1.11 + 1.11i)13-s + (0.862 − 1.70i)14-s + (1.66 + 2.89i)16-s + (−0.911 + 0.244i)17-s + (−0.614 + 0.164i)18-s + (−0.165 − 0.286i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.91690 + 0.415761i\)
\(L(\frac12)\) \(\approx\) \(3.91690 + 0.415761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.543 + 2.58i)T \)
good2 \( 1 + (-2.60 - 0.698i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.69 + 2.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.01 - 4.01i)T - 13iT^{2} \)
17 \( 1 + (3.76 - 1.00i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.721 + 1.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.198 - 0.739i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-3.17 - 1.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.21 - 0.325i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.05iT - 41T^{2} \)
43 \( 1 + (6.74 + 6.74i)T + 43iT^{2} \)
47 \( 1 + (-1.75 + 6.55i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (10.9 - 2.94i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.22 - 5.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.9 + 6.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.145 - 0.544i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 + (1.28 + 4.78i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.79 + 3.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.49 + 1.49i)T - 83iT^{2} \)
89 \( 1 + (1.78 + 3.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.36 + 1.36i)T + 97iT^{2} \)
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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

−11.39566035477684603215335252983, −10.49305766908585102324868809491, −8.806591998794348555869929050131, −7.64747417291642884784178729182, −6.82971250875330388882394508846, −6.43075714765194646663551143156, −5.11319491814690977412922574449, −4.36451940104052176260668782991, −3.32722083012907255967037659092, −1.93776083621549091422013538567, 2.13555947893889559481324075749, 2.98921595218709354621188598359, 4.35917968847266467782308106349, 4.91112372806900500367409274178, 5.83244357943780456535095437576, 6.67371280683345830302022018999, 7.937032689813288694476310182912, 9.525368439215796484289516870771, 10.16367785084660349004359285625, 11.24282438666669275419401158769

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