Properties

Label 2-525-5.4-c3-0-43
Degree $2$
Conductor $525$
Sign $0.894 + 0.447i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.56i·2-s + 3i·3-s − 4.68·4-s − 10.6·6-s − 7i·7-s + 11.8i·8-s − 9·9-s − 5.19·11-s − 14.0i·12-s − 54.5i·13-s + 24.9·14-s − 79.5·16-s + 16.1i·17-s − 32.0i·18-s − 87.4·19-s + ⋯
L(s)  = 1  + 1.25i·2-s + 0.577i·3-s − 0.585·4-s − 0.726·6-s − 0.377i·7-s + 0.521i·8-s − 0.333·9-s − 0.142·11-s − 0.338i·12-s − 1.16i·13-s + 0.475·14-s − 1.24·16-s + 0.230i·17-s − 0.419i·18-s − 1.05·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6224279365\)
\(L(\frac12)\) \(\approx\) \(0.6224279365\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 - 3.56iT - 8T^{2} \)
11 \( 1 + 5.19T + 1.33e3T^{2} \)
13 \( 1 + 54.5iT - 2.19e3T^{2} \)
17 \( 1 - 16.1iT - 4.91e3T^{2} \)
19 \( 1 + 87.4T + 6.85e3T^{2} \)
23 \( 1 + 176. iT - 1.21e4T^{2} \)
29 \( 1 + 142.T + 2.43e4T^{2} \)
31 \( 1 + 94.3T + 2.97e4T^{2} \)
37 \( 1 - 17.3iT - 5.06e4T^{2} \)
41 \( 1 - 210.T + 6.89e4T^{2} \)
43 \( 1 + 521. iT - 7.95e4T^{2} \)
47 \( 1 + 105. iT - 1.03e5T^{2} \)
53 \( 1 - 108. iT - 1.48e5T^{2} \)
59 \( 1 + 210.T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 - 324. iT - 3.00e5T^{2} \)
71 \( 1 - 793.T + 3.57e5T^{2} \)
73 \( 1 + 315. iT - 3.89e5T^{2} \)
79 \( 1 - 425.T + 4.93e5T^{2} \)
83 \( 1 - 283. iT - 5.71e5T^{2} \)
89 \( 1 - 843.T + 7.04e5T^{2} \)
97 \( 1 - 1.53e3iT - 9.12e5T^{2} \)
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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

−10.61680173897015524626678623712, −9.230231509477348081312715052503, −8.395386134598182602306056711834, −7.69504306666785861827085306678, −6.67812544748660144602872388789, −5.81761910568531620050127098119, −4.95847608809309876419895816283, −3.89311467075332569135161413723, −2.41124529685428203914162521764, −0.17998808093967275412602468942, 1.45469342846911195459752545223, 2.25025037778850724628986485304, 3.41908978691637535910624205904, 4.52524891133272392457023782904, 5.93263983812201414376527455143, 6.88395025011908049600612009023, 7.86035935205589560413627603258, 9.169129359276149759340284120216, 9.557507129208766988154447368812, 10.90168445696783595060228021377

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