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Properties

Label 2-525-105.104-c1-0-28
Degree $2$
Conductor $525$
Sign $0.251 + 0.967i$
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  − 2.40·2-s + (1.54 + 0.777i)3-s + 3.79·4-s + (−3.72 − 1.87i)6-s + (−2.44 − i)7-s − 4.31·8-s + (1.79 + 2.40i)9-s − 4.31i·11-s + (5.86 + 2.94i)12-s − 2.44·13-s + (5.89 + 2.40i)14-s + 2.79·16-s − 5.89i·17-s + (−4.31 − 5.79i)18-s − 6.83i·19-s + ⋯
L(s)  = 1  − 1.70·2-s + (0.893 + 0.448i)3-s + 1.89·4-s + (−1.52 − 0.763i)6-s + (−0.925 − 0.377i)7-s − 1.52·8-s + (0.597 + 0.802i)9-s − 1.29i·11-s + (1.69 + 0.850i)12-s − 0.679·13-s + (1.57 + 0.643i)14-s + 0.697·16-s − 1.42i·17-s + (−1.01 − 1.36i)18-s − 1.56i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.251 + 0.967i$
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: \((2,\ 525,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479053 - 0.370532i\)
\(L(\frac12)\) \(\approx\) \(0.479053 - 0.370532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 - 0.777i)T \)
5 \( 1 \)
7 \( 1 + (2.44 + i)T \)
good2 \( 1 + 2.40T + 2T^{2} \)
11 \( 1 + 4.31iT - 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 5.89iT - 17T^{2} \)
19 \( 1 + 6.83iT - 19T^{2} \)
23 \( 1 - 0.502T + 23T^{2} \)
29 \( 1 - 0.502iT - 29T^{2} \)
31 \( 1 - 4.38iT - 31T^{2} \)
37 \( 1 + 0.582iT - 37T^{2} \)
41 \( 1 + 4.66T + 41T^{2} \)
43 \( 1 + 2.58iT - 43T^{2} \)
47 \( 1 + 5.89iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 4.38iT - 61T^{2} \)
67 \( 1 + 14.1iT - 67T^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 - 6.32T + 73T^{2} \)
79 \( 1 - 8.58T + 79T^{2} \)
83 \( 1 - 7.12iT - 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 0.511T + 97T^{2} \)
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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

−10.43546708734475997995083198947, −9.412202845940199910920986812356, −9.173241950378626912387680107029, −8.272013735923046539402742589179, −7.27751958488234538400943731238, −6.71565519923767814478709142871, −5.00602470437950921126272353061, −3.32251106339629723521234378574, −2.49929197559709090496465974826, −0.54401575788286510093822544853, 1.61643969508464119109666710840, 2.53806800813149114098450098161, 3.97659965761204749251212730705, 6.05844737910722438454490478630, 6.94195164938154413802866942084, 7.71086671365071926463862999936, 8.404770792465317561591356163822, 9.329541293555250537168689108341, 9.913170719193058479561921667102, 10.41683295832816381281757820221

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