Properties

Label 2-525-25.11-c1-0-5
Degree $2$
Conductor $525$
Sign $-0.968 - 0.248i$
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  + (0.269 − 0.195i)2-s + (−0.309 + 0.951i)3-s + (−0.583 + 1.79i)4-s + (−0.418 + 2.19i)5-s + (0.102 + 0.316i)6-s − 7-s + (0.399 + 1.22i)8-s + (−0.809 − 0.587i)9-s + (0.316 + 0.672i)10-s + (−0.240 + 0.174i)11-s + (−1.52 − 1.11i)12-s + (0.725 + 0.527i)13-s + (−0.269 + 0.195i)14-s + (−1.95 − 1.07i)15-s + (−2.70 − 1.96i)16-s + (−0.496 − 1.52i)17-s + ⋯
L(s)  = 1  + (0.190 − 0.138i)2-s + (−0.178 + 0.549i)3-s + (−0.291 + 0.898i)4-s + (−0.187 + 0.982i)5-s + (0.0419 + 0.129i)6-s − 0.377·7-s + (0.141 + 0.434i)8-s + (−0.269 − 0.195i)9-s + (0.100 + 0.212i)10-s + (−0.0725 + 0.0527i)11-s + (−0.441 − 0.320i)12-s + (0.201 + 0.146i)13-s + (−0.0718 + 0.0522i)14-s + (−0.505 − 0.278i)15-s + (−0.677 − 0.492i)16-s + (−0.120 − 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112498 + 0.890925i\)
\(L(\frac12)\) \(\approx\) \(0.112498 + 0.890925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.418 - 2.19i)T \)
7 \( 1 + T \)
good2 \( 1 + (-0.269 + 0.195i)T + (0.618 - 1.90i)T^{2} \)
11 \( 1 + (0.240 - 0.174i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.725 - 0.527i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.496 + 1.52i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.214 + 0.661i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.313 - 0.227i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.71 - 5.27i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.911 - 2.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (9.77 + 7.10i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.11 - 4.44i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.36T + 43T^{2} \)
47 \( 1 + (1.04 - 3.20i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.52 - 7.78i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.59 - 6.24i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.67 - 5.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.840 - 2.58i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.01 - 6.18i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.1 + 7.34i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.55 - 4.77i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.96 - 9.12i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.95 - 5.77i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.127 + 0.393i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.15148998969701330544264011089, −10.60625371288172308075299940897, −9.478532782040115648774345354496, −8.723256529899628925002140139701, −7.55795788737447685664995491367, −6.85375279776229748953234740543, −5.63221397883962387345733204506, −4.37567210301751289269074579990, −3.50540164296983998184930076144, −2.61957079908523596113279246137, 0.50293082597889648114814705051, 1.88124882545220526515522620437, 3.83429798389948076103160461826, 4.93576479561763250856292002194, 5.78335566299471966394154272168, 6.56260774733258285298156566972, 7.78626503269117497912108404121, 8.696683466384367485760616266295, 9.533879967679375394495342292456, 10.39996953481178057681811399366

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