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Properties

Label 2-525-35.9-c1-0-12
Degree $2$
Conductor $525$
Sign $0.652 - 0.758i$
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  + (1.73 + i)2-s + (0.866 − 0.5i)3-s + (0.999 + 1.73i)4-s + 1.99·6-s + (0.866 + 2.5i)7-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + (1.73 + i)12-s + i·13-s + (−1.00 + 5.19i)14-s + (1.99 − 3.46i)16-s + (1.73 − 0.999i)18-s + (0.5 − 0.866i)19-s + (2 + 1.73i)21-s + 3.99i·22-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 − 0.288i)3-s + (0.499 + 0.866i)4-s + 0.816·6-s + (0.327 + 0.944i)7-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + (0.499 + 0.288i)12-s + 0.277i·13-s + (−0.267 + 1.38i)14-s + (0.499 − 0.866i)16-s + (0.408 − 0.235i)18-s + (0.114 − 0.198i)19-s + (0.436 + 0.377i)21-s + 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.652 - 0.758i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.652 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91255 + 1.33663i\)
\(L(\frac12)\) \(\approx\) \(2.91255 + 1.33663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 - 2.5i)T \)
good2 \( 1 + (-1.73 - i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.33 + 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (2.59 - 1.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−11.43831192055923908404326556222, −9.879895930559309965807003579510, −9.116219972488702827203435661491, −8.047743300419997876157388592588, −7.16275999646881455723853421380, −6.28311243398311902774125173170, −5.38359078974667623584383529387, −4.44155531981708388385157319805, −3.34748726893631612254052168413, −2.01453804959049159936891766503, 1.64436792264599915532733731249, 3.13903205817527314071090966856, 3.82748842566449692845204616951, 4.76666755198047896407683086719, 5.71428031356590643755939812484, 7.03947466794383415308743400689, 8.083021537177804322227887235269, 9.008709181513731216906854813281, 10.28104141715237831120613397782, 10.81558632308809943994489405482

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