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Properties

Label 2-525-21.5-c1-0-14
Degree $2$
Conductor $525$
Sign $0.909 - 0.415i$
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  + (−0.766 − 0.442i)2-s + (0.189 + 1.72i)3-s + (−0.608 − 1.05i)4-s + (0.616 − 1.40i)6-s + (2.63 + 0.206i)7-s + 2.84i·8-s + (−2.92 + 0.652i)9-s + (1.25 − 0.723i)11-s + (1.69 − 1.24i)12-s − 4.04i·13-s + (−1.93 − 1.32i)14-s + (0.0422 − 0.0730i)16-s + (2.87 + 4.98i)17-s + (2.53 + 0.795i)18-s + (−0.356 − 0.206i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.312i)2-s + (0.109 + 0.993i)3-s + (−0.304 − 0.527i)4-s + (0.251 − 0.572i)6-s + (0.996 + 0.0778i)7-s + 1.00i·8-s + (−0.976 + 0.217i)9-s + (0.377 − 0.218i)11-s + (0.490 − 0.360i)12-s − 1.12i·13-s + (−0.515 − 0.354i)14-s + (0.0105 − 0.0182i)16-s + (0.698 + 1.20i)17-s + (0.596 + 0.187i)18-s + (−0.0818 − 0.0472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10686 + 0.240846i\)
\(L(\frac12)\) \(\approx\) \(1.10686 + 0.240846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.189 - 1.72i)T \)
5 \( 1 \)
7 \( 1 + (-2.63 - 0.206i)T \)
good2 \( 1 + (0.766 + 0.442i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.25 + 0.723i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 + (-2.87 - 4.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.356 + 0.206i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.33 - 3.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.82iT - 29T^{2} \)
31 \( 1 + (-2.30 + 1.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.92 - 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + (5.43 - 9.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.20 - 0.697i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.583 + 1.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.58 - 2.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 5.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + (2.72 - 1.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.42 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (3.90 - 6.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.86iT - 97T^{2} \)
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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

−10.89346626951076718610918051634, −10.07386107761936587810268811900, −9.246261157289159238144418763937, −8.459664403440600472627492157217, −7.80316372205242830482210263567, −5.91906675185487624272272430851, −5.27467363471628459623926701204, −4.30412907424919538605238677096, −2.95130767096222245175227557563, −1.27608703098156438346946464861, 1.00966579435215258755029984030, 2.54846561381199208485637345215, 4.06194523002585878417408607808, 5.21778654103817665421302078554, 6.72599294973414972388021967076, 7.23330861815116851074181291450, 8.065911576251422499565204043353, 8.871766202200855077308259020355, 9.497075591952679711502683805811, 10.96859431722970000652574862881

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