Properties

Label 2-525-25.11-c1-0-14
Degree $2$
Conductor $525$
Sign $0.888 - 0.458i$
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  + (−2.02 + 1.47i)2-s + (−0.309 + 0.951i)3-s + (1.32 − 4.06i)4-s + (2.20 − 0.390i)5-s + (−0.774 − 2.38i)6-s − 7-s + (1.76 + 5.43i)8-s + (−0.809 − 0.587i)9-s + (−3.88 + 4.03i)10-s + (1.92 − 1.39i)11-s + (3.46 + 2.51i)12-s + (−2.89 − 2.10i)13-s + (2.02 − 1.47i)14-s + (−0.309 + 2.21i)15-s + (−4.65 − 3.38i)16-s + (0.830 + 2.55i)17-s + ⋯
L(s)  = 1  + (−1.43 + 1.04i)2-s + (−0.178 + 0.549i)3-s + (0.661 − 2.03i)4-s + (0.984 − 0.174i)5-s + (−0.316 − 0.972i)6-s − 0.377·7-s + (0.623 + 1.92i)8-s + (−0.269 − 0.195i)9-s + (−1.22 + 1.27i)10-s + (0.579 − 0.421i)11-s + (0.999 + 0.726i)12-s + (−0.802 − 0.583i)13-s + (0.541 − 0.393i)14-s + (−0.0798 + 0.571i)15-s + (−1.16 − 0.845i)16-s + (0.201 + 0.619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
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: \((2,\ 525,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687467 + 0.166969i\)
\(L(\frac12)\) \(\approx\) \(0.687467 + 0.166969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 0.390i)T \)
7 \( 1 + T \)
good2 \( 1 + (2.02 - 1.47i)T + (0.618 - 1.90i)T^{2} \)
11 \( 1 + (-1.92 + 1.39i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.89 + 2.10i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.830 - 2.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.88 + 5.81i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.64 + 4.82i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.84 + 5.67i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.810 + 2.49i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.65 - 1.20i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.42 - 1.76i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.78T + 43T^{2} \)
47 \( 1 + (-1.12 + 3.46i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.83 - 8.73i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-12.2 - 8.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.03 - 2.93i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.07 - 6.37i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.49 - 7.66i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.06 + 4.40i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.93 + 15.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.23 + 16.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.43 - 6.85i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.534 + 1.64i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.40201175209037036899090968302, −9.894241748170451132373369198802, −9.026005146120718094014657555206, −8.606269572452573836373272138533, −7.29186107500245954563334076738, −6.41867639431156886798088874522, −5.76813630865291101726983491380, −4.67802660295770425460352896447, −2.61962166237157886135062854389, −0.76401571979112216140134739580, 1.30989421363427196772169530417, 2.23817930406025348482196144213, 3.38751098063189594335005008843, 5.24158908291376617704807221680, 6.68325332606862787180549634714, 7.23684889675053660892827280839, 8.402233184352400526560395379940, 9.475990584758375658600293633473, 9.644889027129911153870462735702, 10.69712945780347110798932888616

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