Properties

Label 2-525-25.16-c1-0-24
Degree $2$
Conductor $525$
Sign $-0.913 + 0.407i$
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  + (0.546 + 0.397i)2-s + (0.309 + 0.951i)3-s + (−0.476 − 1.46i)4-s + (−2.15 + 0.602i)5-s + (−0.208 + 0.642i)6-s − 7-s + (0.739 − 2.27i)8-s + (−0.809 + 0.587i)9-s + (−1.41 − 0.525i)10-s + (−2.81 − 2.04i)11-s + (1.24 − 0.907i)12-s + (−3.95 + 2.87i)13-s + (−0.546 − 0.397i)14-s + (−1.23 − 1.86i)15-s + (−1.18 + 0.863i)16-s + (−2.03 + 6.26i)17-s + ⋯
L(s)  = 1  + (0.386 + 0.280i)2-s + (0.178 + 0.549i)3-s + (−0.238 − 0.733i)4-s + (−0.963 + 0.269i)5-s + (−0.0852 + 0.262i)6-s − 0.377·7-s + (0.261 − 0.805i)8-s + (−0.269 + 0.195i)9-s + (−0.447 − 0.166i)10-s + (−0.849 − 0.617i)11-s + (0.360 − 0.261i)12-s + (−1.09 + 0.797i)13-s + (−0.146 − 0.106i)14-s + (−0.319 − 0.480i)15-s + (−0.297 + 0.215i)16-s + (−0.493 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.913 + 0.407i$
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: \((2,\ 525,\ (\ :1/2),\ -0.913 + 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00552389 - 0.0259133i\)
\(L(\frac12)\) \(\approx\) \(0.00552389 - 0.0259133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.602i)T \)
7 \( 1 + T \)
good2 \( 1 + (-0.546 - 0.397i)T + (0.618 + 1.90i)T^{2} \)
11 \( 1 + (2.81 + 2.04i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.95 - 2.87i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.03 - 6.26i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.575 + 1.77i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (7.47 + 5.43i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.16 - 3.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.17 + 9.78i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.499 - 0.362i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.40 - 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (-2.50 - 7.71i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.826 + 2.54i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.32 + 0.961i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.19 + 1.59i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.63 - 8.11i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.65 - 8.15i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.16 + 3.74i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.09 + 6.44i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.16 - 6.65i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.88 + 2.09i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.96 - 12.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.45275859484492145296155006690, −9.754355573022499454758726794022, −8.671948904100685447018071257649, −7.80307839866601074473136542048, −6.64271662334510013864365469855, −5.81263378774952560178097657865, −4.52498581677690945537713782087, −4.03085987369198800230826099152, −2.52441586142080998583120157173, −0.01253418592694562279925853762, 2.45247575662323658286778427144, 3.33235598281604009615530936430, 4.54306154222022528437147676510, 5.37072578497572034190590797517, 7.14254735841415849207022883677, 7.60768918298973765502628283228, 8.308277515274534118497521775039, 9.413682162135388838586408059175, 10.43407232946921346522504361142, 11.75422419456157155317483482332

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