↑

Properties

Label 2-525-35.34-c2-0-12
Degree $2$
Conductor $525$
Sign $-0.999 + 0.0373i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.91i·2-s + 1.73·3-s − 4.51·4-s + 5.05i·6-s + (3.36 + 6.13i)7-s − 1.49i·8-s + 2.99·9-s − 2.58·11-s − 7.81·12-s + 0.0498·13-s + (−17.9 + 9.80i)14-s − 13.6·16-s + 14.2·17-s + 8.75i·18-s + 14.9i·19-s + ⋯
L(s)  = 1  + 1.45i·2-s + 0.577·3-s − 1.12·4-s + 0.842i·6-s + (0.480 + 0.877i)7-s − 0.186i·8-s + 0.333·9-s − 0.235·11-s − 0.651·12-s + 0.00383·13-s + (−1.27 + 0.700i)14-s − 0.855·16-s + 0.835·17-s + 0.486i·18-s + 0.785i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0373i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 + 0.0373i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.999 + 0.0373i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.000727020\)
\(L(\frac12)\) \(\approx\) \(2.000727020\)
\(L(2)\) 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 \( 1 + (-3.36 - 6.13i)T \)
good2 \( 1 - 2.91iT - 4T^{2} \)
11 \( 1 + 2.58T + 121T^{2} \)
13 \( 1 - 0.0498T + 169T^{2} \)
17 \( 1 - 14.2T + 289T^{2} \)
19 \( 1 - 14.9iT - 361T^{2} \)
23 \( 1 - 22.2iT - 529T^{2} \)
29 \( 1 + 17.4T + 841T^{2} \)
31 \( 1 - 6.36iT - 961T^{2} \)
37 \( 1 + 7.14iT - 1.36e3T^{2} \)
41 \( 1 - 74.5iT - 1.68e3T^{2} \)
43 \( 1 + 79.2iT - 1.84e3T^{2} \)
47 \( 1 + 81.3T + 2.20e3T^{2} \)
53 \( 1 - 67.1iT - 2.80e3T^{2} \)
59 \( 1 + 4.33iT - 3.48e3T^{2} \)
61 \( 1 + 109. iT - 3.72e3T^{2} \)
67 \( 1 + 49.1iT - 4.48e3T^{2} \)
71 \( 1 - 97.3T + 5.04e3T^{2} \)
73 \( 1 - 116.T + 5.32e3T^{2} \)
79 \( 1 + 98.8T + 6.24e3T^{2} \)
83 \( 1 - 112.T + 6.88e3T^{2} \)
89 \( 1 - 77.7iT - 7.92e3T^{2} \)
97 \( 1 - 109.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10958036556255609033071259655, −9.779722518916714051381258954085, −9.049276955040283751180687698602, −8.026253515250147579493652070633, −7.78842842999993230573384776986, −6.55786901206730958054241682046, −5.61954380263356482836615245059, −4.91059180650183553286903983789, −3.42664059541643540354291785770, −1.92025134452261917934514432415, 0.74308389930456965461877752565, 1.99187421001700082476843836839, 3.12240406804953722201009744008, 4.04769617154673325509731296345, 4.99417515534535445801307996894, 6.70836773997521260679183886086, 7.69349166408545608199073402796, 8.644543634991386195058420406935, 9.655816549245952543032573958040, 10.29288384662963179846631822956

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