↑

Properties

Label 2-525-35.34-c2-0-4
Degree $2$
Conductor $525$
Sign $-0.818 + 0.574i$
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  + 2.67i·2-s − 1.73·3-s − 3.18·4-s − 4.64i·6-s + (3.32 − 6.16i)7-s + 2.19i·8-s + 2.99·9-s + 13.6·11-s + 5.50·12-s − 24.7·13-s + (16.5 + 8.90i)14-s − 18.6·16-s − 11.4·17-s + 8.03i·18-s + 33.2i·19-s + ⋯
L(s)  = 1  + 1.33i·2-s − 0.577·3-s − 0.795·4-s − 0.773i·6-s + (0.474 − 0.880i)7-s + 0.274i·8-s + 0.333·9-s + 1.23·11-s + 0.459·12-s − 1.90·13-s + (1.17 + 0.635i)14-s − 1.16·16-s − 0.671·17-s + 0.446i·18-s + 1.74i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.818 + 0.574i$
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: \((2,\ 525,\ (\ :1),\ -0.818 + 0.574i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7565217119\)
\(L(\frac12)\) \(\approx\) \(0.7565217119\)
\(L(2)\) 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 \( 1 + (-3.32 + 6.16i)T \)
good2 \( 1 - 2.67iT - 4T^{2} \)
11 \( 1 - 13.6T + 121T^{2} \)
13 \( 1 + 24.7T + 169T^{2} \)
17 \( 1 + 11.4T + 289T^{2} \)
19 \( 1 - 33.2iT - 361T^{2} \)
23 \( 1 - 34.7iT - 529T^{2} \)
29 \( 1 + 26.5T + 841T^{2} \)
31 \( 1 + 18.5iT - 961T^{2} \)
37 \( 1 - 24.5iT - 1.36e3T^{2} \)
41 \( 1 + 0.626iT - 1.68e3T^{2} \)
43 \( 1 - 19.8iT - 1.84e3T^{2} \)
47 \( 1 + 41.9T + 2.20e3T^{2} \)
53 \( 1 + 24.7iT - 2.80e3T^{2} \)
59 \( 1 + 73.3iT - 3.48e3T^{2} \)
61 \( 1 - 71.9iT - 3.72e3T^{2} \)
67 \( 1 - 58.7iT - 4.48e3T^{2} \)
71 \( 1 - 82.1T + 5.04e3T^{2} \)
73 \( 1 + 97.7T + 5.32e3T^{2} \)
79 \( 1 + 69.3T + 6.24e3T^{2} \)
83 \( 1 + 27.2T + 6.88e3T^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 - 178.T + 9.40e3T^{2} \)
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

−11.39038570439355802150778859553, −10.07744888014782616312529894167, −9.400244381492539430108119865244, −8.064727061287534139446710803318, −7.40196548582597136025530823254, −6.75174662585247577778673915573, −5.76410788426273905739660046806, −4.86197162875335858759960263783, −3.92539967573756741833743647708, −1.72517493943748805474436117668, 0.30698043506121295815021812638, 1.92650251913919211236187759466, 2.77178268932566694657168055294, 4.36397376973718041389299792627, 4.99208270217748088794683574122, 6.47384717951846136368244860273, 7.23129015577549317105227524561, 8.901015110352039340854998662432, 9.333016960090533280088679266236, 10.35172086755394632190455843064

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