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Properties

Label 2-525-21.17-c1-0-24
Degree $2$
Conductor $525$
Sign $0.575 - 0.817i$
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  + (−1.31 + 0.757i)2-s + (1.68 + 0.419i)3-s + (0.147 − 0.254i)4-s + (−2.52 + 0.722i)6-s + (2.64 − 0.0753i)7-s − 2.58i·8-s + (2.64 + 1.41i)9-s + (1.86 + 1.07i)11-s + (0.354 − 0.366i)12-s − 3.48i·13-s + (−3.41 + 2.10i)14-s + (2.25 + 3.89i)16-s + (1.78 − 3.09i)17-s + (−4.54 + 0.155i)18-s + (1.05 − 0.611i)19-s + ⋯
L(s)  = 1  + (−0.927 + 0.535i)2-s + (0.970 + 0.242i)3-s + (0.0735 − 0.127i)4-s + (−1.02 + 0.294i)6-s + (0.999 − 0.0284i)7-s − 0.913i·8-s + (0.882 + 0.470i)9-s + (0.560 + 0.323i)11-s + (0.102 − 0.105i)12-s − 0.965i·13-s + (−0.911 + 0.561i)14-s + (0.562 + 0.974i)16-s + (0.433 − 0.751i)17-s + (−1.07 + 0.0366i)18-s + (0.242 − 0.140i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22699 + 0.637027i\)
\(L(\frac12)\) \(\approx\) \(1.22699 + 0.637027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.419i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.0753i)T \)
good2 \( 1 + (1.31 - 0.757i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.86 - 1.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.48iT - 13T^{2} \)
17 \( 1 + (-1.78 + 3.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.05 + 0.611i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.31 - 0.757i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-2.75 - 1.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.90 + 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 2.99T + 43T^{2} \)
47 \( 1 + (-3.05 - 5.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.72 - 5.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.08 + 1.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.94 - 1.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.15 - 8.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (5.93 + 3.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.941 + 1.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.10T + 83T^{2} \)
89 \( 1 + (0.889 + 1.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.32iT - 97T^{2} \)
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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

−10.51773462965515936141110968366, −9.932936092646509075130036262821, −8.908781666697503736647970092032, −8.472778269753976936980511544409, −7.51826124239390937328590131026, −7.06352052081798874775390596332, −5.36773005265835779437225296702, −4.21808012100605119575226880058, −3.07868810136765849737473100135, −1.38817168355848143595041369169, 1.35320811118167663033591337319, 2.17241632149764651055055524402, 3.70529374098001246334232092434, 4.88323611923451885495215417935, 6.32425688229842505439013127067, 7.55008040233622765030548210217, 8.431105002231389303208012246840, 8.770239657825963669549337011282, 9.837105899729874657588135200494, 10.41237792628186664250728447321

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