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Properties

Label 2-525-105.59-c1-0-17
Degree $2$
Conductor $525$
Sign $-0.821 - 0.569i$
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 + (1.35 + 1.08i)3-s + (−1.12 + 1.95i)4-s + (−0.543 + 3.53i)6-s + (−2.64 + 0.00953i)7-s − 0.527·8-s + (0.649 + 2.92i)9-s + (4.06 + 2.34i)11-s + (−3.64 + 1.41i)12-s − 0.638·13-s + (−2.74 − 4.71i)14-s + (1.71 + 2.96i)16-s + (−3.59 − 2.07i)17-s + (−4.56 + 4.18i)18-s + (0.776 − 0.448i)19-s + ⋯
L(s)  = 1  + (0.729 + 1.26i)2-s + (0.779 + 0.625i)3-s + (−0.563 + 0.976i)4-s + (−0.221 + 1.44i)6-s + (−0.999 + 0.00360i)7-s − 0.186·8-s + (0.216 + 0.976i)9-s + (1.22 + 0.707i)11-s + (−1.05 + 0.408i)12-s − 0.177·13-s + (−0.733 − 1.26i)14-s + (0.427 + 0.741i)16-s + (−0.871 − 0.503i)17-s + (−1.07 + 0.985i)18-s + (0.178 − 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.821 - 0.569i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.821 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756937 + 2.42153i\)
\(L(\frac12)\) \(\approx\) \(0.756937 + 2.42153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.35 - 1.08i)T \)
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

−11.16743650617232257508632200058, −9.981947892463918148668052918917, −9.344527263067732596052159649303, −8.462879352894653074390922595585, −7.34547879644072285225580997212, −6.69902763602192711279536918018, −5.74596993843745128083457688271, −4.39241165822148760130187501355, −4.00901252772943624804372465655, −2.47249016020762272991708547114, 1.24517209110766941346266573143, 2.52121174919344099639509690976, 3.51975538889782579062091010000, 4.11674440425182813473004852502, 5.86175906719058988835179696474, 6.73639259605973117612068813365, 7.84265849084843077019130168977, 9.105684975715837592203041082220, 9.575779880864718593821099683346, 10.68693078119313272432364068529

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