Properties

Label 2-525-25.16-c1-0-17
Degree $2$
Conductor $525$
Sign $0.794 - 0.607i$
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  + (1.16 + 0.847i)2-s + (0.309 + 0.951i)3-s + (0.0240 + 0.0741i)4-s + (2.06 − 0.848i)5-s + (−0.445 + 1.37i)6-s + 7-s + (0.856 − 2.63i)8-s + (−0.809 + 0.587i)9-s + (3.13 + 0.762i)10-s + (0.597 + 0.434i)11-s + (−0.0630 + 0.0458i)12-s + (0.00453 − 0.00329i)13-s + (1.16 + 0.847i)14-s + (1.44 + 1.70i)15-s + (3.35 − 2.43i)16-s + (−1.18 + 3.63i)17-s + ⋯
L(s)  = 1  + (0.824 + 0.599i)2-s + (0.178 + 0.549i)3-s + (0.0120 + 0.0370i)4-s + (0.925 − 0.379i)5-s + (−0.181 + 0.559i)6-s + 0.377·7-s + (0.302 − 0.931i)8-s + (−0.269 + 0.195i)9-s + (0.990 + 0.241i)10-s + (0.180 + 0.130i)11-s + (−0.0182 + 0.0132i)12-s + (0.00125 − 0.000914i)13-s + (0.311 + 0.226i)14-s + (0.373 + 0.440i)15-s + (0.839 − 0.609i)16-s + (−0.286 + 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53604 + 0.859069i\)
\(L(\frac12)\) \(\approx\) \(2.53604 + 0.859069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-2.06 + 0.848i)T \)
7 \( 1 - T \)
good2 \( 1 + (-1.16 - 0.847i)T + (0.618 + 1.90i)T^{2} \)
11 \( 1 + (-0.597 - 0.434i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.00453 + 0.00329i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.18 - 3.63i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.983 + 3.02i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.00 + 0.733i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.00 - 3.09i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.15 - 6.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.02 - 4.37i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.453 - 0.329i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.55T + 43T^{2} \)
47 \( 1 + (-1.02 - 3.16i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.55 + 4.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.34 + 3.15i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.09 + 3.70i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.73 - 8.41i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.02 + 6.23i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.53 - 4.75i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.25 + 16.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.48 + 10.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.49 + 1.81i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.21 - 16.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.68964119139981332449859996666, −10.11895998414998659045936885414, −9.172809413354800576903606316244, −8.399764404030873050028304510406, −6.97797161264711552272087695190, −6.17916951583243979398522920886, −5.15329231102666876496709864108, −4.66770216876090269229596151445, −3.38341620397902636826765936494, −1.64400955560031872773569551882, 1.77093040098839965270617759462, 2.69010559913122883074472996508, 3.82506994255725637943282791153, 5.11677397229995109071333386669, 5.93118962376340792477591627726, 7.08594388703858024033907793620, 8.031930126769884089780810591498, 9.031012750009855436824622667790, 10.03435878548227528163161437187, 11.05432819361015632258415702796

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