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Properties

Label 2-525-5.4-c1-0-9
Degree $2$
Conductor $525$
Sign $0.894 + 0.447i$
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  + 0.381i·2-s − i·3-s + 1.85·4-s + 0.381·6-s − i·7-s + 1.47i·8-s − 9-s + 3.47·11-s − 1.85i·12-s − 5.23i·13-s + 0.381·14-s + 3.14·16-s + 5.70i·17-s − 0.381i·18-s − 1.23·19-s + ⋯
L(s)  = 1  + 0.270i·2-s − 0.577i·3-s + 0.927·4-s + 0.155·6-s − 0.377i·7-s + 0.520i·8-s − 0.333·9-s + 1.04·11-s − 0.535i·12-s − 1.45i·13-s + 0.102·14-s + 0.786·16-s + 1.38i·17-s − 0.0900i·18-s − 0.283·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79594 - 0.423964i\)
\(L(\frac12)\) \(\approx\) \(1.79594 - 0.423964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
7 \( 1 + iT \)
good2 \( 1 - 0.381iT - 2T^{2} \)
11 \( 1 - 3.47T + 11T^{2} \)
13 \( 1 + 5.23iT - 13T^{2} \)
17 \( 1 - 5.70iT - 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 - 8.70T + 29T^{2} \)
31 \( 1 + 4.47T + 31T^{2} \)
37 \( 1 + 3.47iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 3.76iT - 43T^{2} \)
47 \( 1 - 2.76iT - 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 - 5.23T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 - 3.23iT - 73T^{2} \)
79 \( 1 + 6.23T + 79T^{2} \)
83 \( 1 + 3.52iT - 83T^{2} \)
89 \( 1 + 7.70T + 89T^{2} \)
97 \( 1 - 3.52iT - 97T^{2} \)
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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

−10.66884442467750785423484458494, −10.27547435197567621944367647000, −8.645046033407804014125524944149, −8.069571136790590178651624923000, −7.02135852238868284906466194689, −6.38209003437364867952162510224, −5.51662501056955730919185238440, −3.91637536216849347108536082304, −2.66502739460181134897732335368, −1.27679661244757547946785259237, 1.68028554892747885178977757722, 2.97834451304383027415082022637, 4.08066378282328285310519425593, 5.25222771584494571758207954451, 6.53698472330534938503928165471, 7.01504206603295341141920793164, 8.430181683853006088593644166796, 9.391425554308781922399871278380, 9.938406012142197214258160286360, 11.16833541413012396152272091550

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