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Properties

Label 2-525-35.9-c1-0-15
Degree $2$
Conductor $525$
Sign $-0.989 - 0.147i$
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  + (−2.34 − 1.35i)2-s + (0.866 − 0.5i)3-s + (2.65 + 4.59i)4-s − 2.70·6-s + (−2.06 − 1.65i)7-s − 8.93i·8-s + (0.499 − 0.866i)9-s + (−1.41 − 2.44i)11-s + (4.59 + 2.65i)12-s + 4.47i·13-s + (2.60 + 6.66i)14-s + (−6.76 + 11.7i)16-s + (3.66 − 2.11i)17-s + (−2.34 + 1.35i)18-s + (1.05 − 1.81i)19-s + ⋯
L(s)  = 1  + (−1.65 − 0.955i)2-s + (0.499 − 0.288i)3-s + (1.32 + 2.29i)4-s − 1.10·6-s + (−0.780 − 0.624i)7-s − 3.15i·8-s + (0.166 − 0.288i)9-s + (−0.425 − 0.737i)11-s + (1.32 + 0.765i)12-s + 1.24i·13-s + (0.695 + 1.78i)14-s + (−1.69 + 2.93i)16-s + (0.888 − 0.513i)17-s + (−0.551 + 0.318i)18-s + (0.240 − 0.417i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0293621 + 0.396922i\)
\(L(\frac12)\) \(\approx\) \(0.0293621 + 0.396922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (2.06 + 1.65i)T \)
good2 \( 1 + (2.34 + 1.35i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1.41 + 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 + (-3.66 + 2.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.05 + 1.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.72 + 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (4.66 + 8.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.64 + 2.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.37T + 41T^{2} \)
43 \( 1 - 3.13iT - 43T^{2} \)
47 \( 1 + (6.74 + 3.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.23 - 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.89 + 3.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.86 - 1.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (2.29 - 1.32i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.02iT - 83T^{2} \)
89 \( 1 + (4.11 - 7.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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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

−10.15612217223373492620220092943, −9.533823536673910923391461336165, −8.881418241563634668175770054839, −7.85512406886720740718990723030, −7.28076897816513004454666232405, −6.25983666546282407401229637354, −3.96031110520012125640423714549, −3.06924885625310435143392763558, −1.91161374741507390954736468404, −0.36048736014284744019444323368, 1.77185203113840003604793935868, 3.22794506943390851775068133434, 5.34557319203000121215811633209, 5.94787055032171191916028817383, 7.19940611513617081614637178681, 7.86937321418318221828120030150, 8.599153943436103316962315404925, 9.530410415642439814241293099872, 10.07509313473769310368095183938, 10.62572721400725103770192767082

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