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Properties

Label 2-525-105.89-c1-0-36
Degree $2$
Conductor $525$
Sign $-0.336 + 0.941i$
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  + (1.03 − 1.78i)2-s + (1.61 − 0.627i)3-s + (−1.12 − 1.95i)4-s + (0.543 − 3.53i)6-s + (2.64 + 0.00953i)7-s − 0.527·8-s + (2.21 − 2.02i)9-s + (−4.06 + 2.34i)11-s + (−3.04 − 2.44i)12-s + 0.638·13-s + (2.74 − 4.71i)14-s + (1.71 − 2.96i)16-s + (−3.59 + 2.07i)17-s + (−1.33 − 6.04i)18-s + (0.776 + 0.448i)19-s + ⋯
L(s)  = 1  + (0.729 − 1.26i)2-s + (0.932 − 0.362i)3-s + (−0.563 − 0.976i)4-s + (0.221 − 1.44i)6-s + (0.999 + 0.00360i)7-s − 0.186·8-s + (0.737 − 0.675i)9-s + (−1.22 + 0.707i)11-s + (−0.879 − 0.705i)12-s + 0.177·13-s + (0.733 − 1.26i)14-s + (0.427 − 0.741i)16-s + (−0.871 + 0.503i)17-s + (−0.315 − 1.42i)18-s + (0.178 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69628 - 2.40706i\)
\(L(\frac12)\) \(\approx\) \(1.69628 - 2.40706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 + 0.627i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.00953i)T \)
good2 \( 1 + (-1.03 + 1.78i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (4.06 - 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.638T + 13T^{2} \)
17 \( 1 + (3.59 - 2.07i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.776 - 0.448i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.40 - 5.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 + (2.02 - 1.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.85 + 5.69i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 - 3.14iT - 43T^{2} \)
47 \( 1 + (-5.89 - 3.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.13 - 1.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.254 - 0.440i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.48 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.18 - 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.22iT - 71T^{2} \)
73 \( 1 + (-7.22 - 12.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.54 + 7.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.76iT - 83T^{2} \)
89 \( 1 + (-6.90 + 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.66628862709908928677352057324, −10.01980380747678933897912452186, −8.898504096390260073301531706916, −7.894791188633014426024634819655, −7.25451401090268461152857951435, −5.52055298857536417045725935417, −4.51035794026246126231600491678, −3.61384043744075706760091993724, −2.35582075671386229767245906980, −1.68127240916617916848456850759, 2.21814326971787094854441094236, 3.65380986019976617002236462637, 4.78608340412480313478497225770, 5.29096873673981054312993938106, 6.63845073891463127751520715869, 7.58279707159082500716152383030, 8.293176084868042512549508276179, 8.816494949077891987711365123239, 10.35810726602816197331201784179, 10.87802430344186592854341600378

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