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Properties

Label 2-525-105.104-c1-0-26
Degree $2$
Conductor $525$
Sign $-0.111 + 0.993i$
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  − 1.09·2-s + (0.323 − 1.70i)3-s − 0.791·4-s + (−0.355 + 1.87i)6-s + (2.44 + i)7-s + 3.06·8-s + (−2.79 − 1.09i)9-s − 3.06i·11-s + (−0.255 + 1.34i)12-s + 2.44·13-s + (−2.69 − 1.09i)14-s − 1.79·16-s − 2.69i·17-s + (3.06 + 1.20i)18-s + 4.38i·19-s + ⋯
L(s)  = 1  − 0.777·2-s + (0.186 − 0.982i)3-s − 0.395·4-s + (−0.144 + 0.763i)6-s + (0.925 + 0.377i)7-s + 1.08·8-s + (−0.930 − 0.366i)9-s − 0.925i·11-s + (−0.0737 + 0.388i)12-s + 0.679·13-s + (−0.719 − 0.293i)14-s − 0.447·16-s − 0.653i·17-s + (0.723 + 0.284i)18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.111 + 0.993i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616153 - 0.689029i\)
\(L(\frac12)\) \(\approx\) \(0.616153 - 0.689029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.323 + 1.70i)T \)
5 \( 1 \)
7 \( 1 + (-2.44 - i)T \)
good2 \( 1 + 1.09T + 2T^{2} \)
11 \( 1 + 3.06iT - 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 2.69iT - 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 - 5.26T + 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 + 6.83iT - 31T^{2} \)
37 \( 1 + 8.58iT - 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 6.58iT - 43T^{2} \)
47 \( 1 + 2.69iT - 47T^{2} \)
53 \( 1 + 3.93T + 53T^{2} \)
59 \( 1 + 7.51T + 59T^{2} \)
61 \( 1 - 6.83iT - 61T^{2} \)
67 \( 1 + 4.16iT - 67T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 - 16.1T + 73T^{2} \)
79 \( 1 + 0.582T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 7.51T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−10.72883620635928795584609752230, −9.410283173207150914955909502677, −8.674701637055155110648863285949, −8.093348190401457623250421818036, −7.38141699889761127649224929093, −6.07641155210349842967616788979, −5.16044681187479863854667454203, −3.66208082941647762765850933882, −2.04930744365160506444409857172, −0.790496798338438195516719997049, 1.50703478872330246170754156174, 3.38110662507074087999899644331, 4.72261686177246790873297109172, 4.94136395789514421132196476977, 6.76126748113426243016770535431, 7.88469856198650617324458870271, 8.630942104137068477181844126139, 9.230539670301388692039409617633, 10.24016913604915963668814315653, 10.74291524081581183741463249124

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