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Properties

Label 2-525-105.89-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.329 + 0.944i$
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  + (−1.36 + 2.36i)2-s + (−1.70 − 0.310i)3-s + (−2.73 − 4.74i)4-s + (3.06 − 3.61i)6-s + (1.39 + 2.24i)7-s + 9.49·8-s + (2.80 + 1.05i)9-s + (−1.79 + 1.03i)11-s + (3.19 + 8.92i)12-s − 2.04·13-s + (−7.22 + 0.226i)14-s + (−7.50 + 13.0i)16-s + (−1.82 + 1.05i)17-s + (−6.34 + 5.20i)18-s + (2.41 + 1.39i)19-s + ⋯
L(s)  = 1  + (−0.966 + 1.67i)2-s + (−0.983 − 0.179i)3-s + (−1.36 − 2.37i)4-s + (1.25 − 1.47i)6-s + (0.526 + 0.849i)7-s + 3.35·8-s + (0.935 + 0.352i)9-s + (−0.540 + 0.312i)11-s + (0.921 + 2.57i)12-s − 0.566·13-s + (−1.93 + 0.0604i)14-s + (−1.87 + 3.25i)16-s + (−0.441 + 0.255i)17-s + (−1.49 + 1.22i)18-s + (0.553 + 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114331 - 0.160945i\)
\(L(\frac12)\) \(\approx\) \(0.114331 - 0.160945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.310i)T \)
5 \( 1 \)
7 \( 1 + (-1.39 - 2.24i)T \)
good2 \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.79 - 1.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.04T + 13T^{2} \)
17 \( 1 + (1.82 - 1.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.41 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.888 - 1.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (-6.04 + 3.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.25 + 3.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 4.37iT - 43T^{2} \)
47 \( 1 + (2.55 + 1.47i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.09 + 5.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.44 + 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.30 + 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.41 - 3.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + (3.35 + 5.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.95 - 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.96iT - 83T^{2} \)
89 \( 1 + (6.20 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.97T + 97T^{2} \)
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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

−11.23967785350691136469691841966, −10.23644671978500830574113149665, −9.634607339097565995364575487110, −8.533932000814653223424108720164, −7.79294877711202877701030509087, −6.97797748113118477102282500423, −6.09969731739375471171662194708, −5.28023250698677544503953918418, −4.70211472452783632744760820654, −1.68547364917703087615586272700, 0.19315715188234720552864565918, 1.50997712679737247112500513392, 3.04378552329864906381263762848, 4.31077317092268468526961136838, 5.02786956131667910269003821500, 6.90489389232815638050360309291, 7.78354748643258507867859275337, 8.738627161937545551929652942842, 9.873063105246719864130066534927, 10.35051801444276353065310546362

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