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Properties

Label 2-525-105.59-c1-0-41
Degree $2$
Conductor $525$
Sign $-0.701 - 0.712i$
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  + (−0.442 − 0.766i)2-s + (−1.02 − 1.39i)3-s + (0.608 − 1.05i)4-s + (−0.616 + 1.40i)6-s + (−0.206 − 2.63i)7-s − 2.84·8-s + (−0.899 + 2.86i)9-s + (−1.25 − 0.723i)11-s + (−2.09 + 0.230i)12-s − 4.04·13-s + (−1.93 + 1.32i)14-s + (0.0422 + 0.0730i)16-s + (4.98 + 2.87i)17-s + (2.59 − 0.577i)18-s + (0.356 − 0.206i)19-s + ⋯
L(s)  = 1  + (−0.312 − 0.541i)2-s + (−0.591 − 0.806i)3-s + (0.304 − 0.527i)4-s + (−0.251 + 0.572i)6-s + (−0.0778 − 0.996i)7-s − 1.00·8-s + (−0.299 + 0.954i)9-s + (−0.377 − 0.218i)11-s + (−0.604 + 0.0665i)12-s − 1.12·13-s + (−0.515 + 0.354i)14-s + (0.0105 + 0.0182i)16-s + (1.20 + 0.698i)17-s + (0.610 − 0.136i)18-s + (0.0818 − 0.0472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221843 + 0.530012i\)
\(L(\frac12)\) \(\approx\) \(0.221843 + 0.530012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.02 + 1.39i)T \)
5 \( 1 \)
7 \( 1 + (0.206 + 2.63i)T \)
good2 \( 1 + (0.442 + 0.766i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.25 + 0.723i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.04T + 13T^{2} \)
17 \( 1 + (-4.98 - 2.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.356 + 0.206i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.65 + 6.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.82iT - 29T^{2} \)
31 \( 1 + (-2.30 - 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.06 - 2.92i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.46T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (-9.41 + 5.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.697 - 1.20i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (5.99 + 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + (-1.57 + 2.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.42 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (3.90 + 6.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.86T + 97T^{2} \)
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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

−10.29181098076680423209796844074, −9.964989846918627640178017560375, −8.400640792976865626680323995634, −7.52252335650263365787120080346, −6.63551607415974621029977559324, −5.79170227349852222447570822778, −4.70432384109376406074858702646, −2.98605251202032628347291827578, −1.69594981234805983442852559808, −0.38308372254170213932427192781, 2.58842722516549609834559223729, 3.67186661723803132256409476321, 5.23285050260235664305780235529, 5.69777483763326963078243362230, 6.95529345141047711797300513310, 7.76028995674317121107440013798, 8.847782103518877975968907293807, 9.589388971283383815733467676221, 10.31646515858480039829877822326, 11.77015349267490452117056612440

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