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Properties

Label 2-525-25.11-c1-0-23
Degree $2$
Conductor $525$
Sign $-0.466 + 0.884i$
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  + (1.98 − 1.44i)2-s + (−0.309 + 0.951i)3-s + (1.24 − 3.84i)4-s + (−1.03 − 1.98i)5-s + (0.759 + 2.33i)6-s − 7-s + (−1.55 − 4.77i)8-s + (−0.809 − 0.587i)9-s + (−4.91 − 2.45i)10-s + (4.88 − 3.54i)11-s + (3.26 + 2.37i)12-s + (−1.76 − 1.28i)13-s + (−1.98 + 1.44i)14-s + (2.20 − 0.369i)15-s + (−3.43 − 2.49i)16-s + (1.15 + 3.54i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.02i)2-s + (−0.178 + 0.549i)3-s + (0.624 − 1.92i)4-s + (−0.461 − 0.886i)5-s + (0.310 + 0.954i)6-s − 0.377·7-s + (−0.548 − 1.68i)8-s + (−0.269 − 0.195i)9-s + (−1.55 − 0.775i)10-s + (1.47 − 1.06i)11-s + (0.943 + 0.685i)12-s + (−0.490 − 0.356i)13-s + (−0.531 + 0.386i)14-s + (0.569 − 0.0953i)15-s + (−0.858 − 0.624i)16-s + (0.279 + 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.466 + 0.884i$
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: \((2,\ 525,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33739 - 2.21666i\)
\(L(\frac12)\) \(\approx\) \(1.33739 - 2.21666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 + 1.98i)T \)
7 \( 1 + T \)
good2 \( 1 + (-1.98 + 1.44i)T + (0.618 - 1.90i)T^{2} \)
11 \( 1 + (-4.88 + 3.54i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.76 + 1.28i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.15 - 3.54i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.28 + 3.95i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.91 - 3.57i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.87 + 5.76i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.336 - 1.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.65 - 2.65i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.93 - 6.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (2.93 - 9.04i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.558 - 1.71i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.542 - 0.394i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.77 - 2.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.22 + 6.84i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.21 - 6.80i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.536 - 0.389i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.51 - 10.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.115 - 0.354i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.71 - 4.15i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.91 + 15.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.04986912427850551324742754162, −9.855763480472165815775556673732, −9.165552801392659610894339119550, −7.992251582374346263419729420364, −6.22126645476811920460532834203, −5.71380856909325305306441288014, −4.38355601297737921374145123917, −4.01864071575918161505300361174, −2.89326018672906372073560411271, −1.09621870776059295638894716767, 2.41209528464301465142309497437, 3.75987476850924803450930867467, 4.45012098531853044728400630230, 5.83714042277685431217281460857, 6.59818016971318691741935315158, 7.15811079523655621186705482704, 7.80767475164327832186974884604, 9.243503174621144897859148871035, 10.42247591635330691810331546846, 11.78508136056609820477482878998

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