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Properties

Label 2-525-105.104-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.129 - 0.991i$
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  − 1.73·3-s − 2·4-s + (0.866 − 2.5i)7-s + 2.99·9-s + 3.46·12-s − 5.19·13-s + 4·16-s + 8.66i·19-s + (−1.49 + 4.33i)21-s − 5.19·27-s + (−1.73 + 5i)28-s + 8.66i·31-s − 5.99·36-s + 10i·37-s + 9·39-s + ⋯
L(s)  = 1  − 1.00·3-s − 4-s + (0.327 − 0.944i)7-s + 0.999·9-s + 1.00·12-s − 1.44·13-s + 16-s + 1.98i·19-s + (−0.327 + 0.944i)21-s − 1.00·27-s + (−0.327 + 0.944i)28-s + 1.55i·31-s − 0.999·36-s + 1.64i·37-s + 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.129 - 0.991i$
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: \((2,\ 525,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289103 + 0.329418i\)
\(L(\frac12)\) \(\approx\) \(0.289103 + 0.329418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 \)
7 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + 2T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−10.97597614297880465032724013415, −10.02746380944785580204169477457, −9.815122855065261890362439773622, −8.273361313864786450872978086708, −7.52691468897359460854277318247, −6.48180853456505477535957617944, −5.25319537156480419360196590242, −4.65708942210780292188596659207, −3.63816324703329082798230012468, −1.32405775916972774205726664311, 0.33090966372630186139671030315, 2.40628662692044119453696030807, 4.23950650292507199390242448515, 5.05521707132237474578327744846, 5.65537697477364257038411022252, 6.94528768279055183720331696645, 7.892373657994132158383364893483, 9.170377392242587473362865031441, 9.542994875702193078198259682580, 10.69065134712188122127316533885

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