↑

Properties

Label 2-525-15.14-c2-0-17
Degree $2$
Conductor $525$
Sign $-0.438 - 0.898i$
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  − 1.30·2-s + (2.38 + 1.82i)3-s − 2.29·4-s + (−3.11 − 2.38i)6-s − 2.64i·7-s + 8.22·8-s + (2.35 + 8.68i)9-s + 2.61i·11-s + (−5.45 − 4.17i)12-s + 6.35i·13-s + 3.45i·14-s − 1.58·16-s − 12.1·17-s + (−3.07 − 11.3i)18-s + 10.2·19-s + ⋯
L(s)  = 1  − 0.653·2-s + (0.794 + 0.607i)3-s − 0.572·4-s + (−0.519 − 0.397i)6-s − 0.377i·7-s + 1.02·8-s + (0.261 + 0.965i)9-s + 0.237i·11-s + (−0.454 − 0.348i)12-s + 0.488i·13-s + 0.247i·14-s − 0.0989·16-s − 0.714·17-s + (−0.170 − 0.630i)18-s + 0.538·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.438 - 0.898i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.438 - 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.077607346\)
\(L(\frac12)\) \(\approx\) \(1.077607346\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.38 - 1.82i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good2 \( 1 + 1.30T + 4T^{2} \)
11 \( 1 - 2.61iT - 121T^{2} \)
13 \( 1 - 6.35iT - 169T^{2} \)
17 \( 1 + 12.1T + 289T^{2} \)
19 \( 1 - 10.2T + 361T^{2} \)
23 \( 1 + 4.30T + 529T^{2} \)
29 \( 1 - 17.3iT - 841T^{2} \)
31 \( 1 - 39.2T + 961T^{2} \)
37 \( 1 - 41.0iT - 1.36e3T^{2} \)
41 \( 1 - 30.2iT - 1.68e3T^{2} \)
43 \( 1 - 55.8iT - 1.84e3T^{2} \)
47 \( 1 + 39.9T + 2.20e3T^{2} \)
53 \( 1 + 105.T + 2.80e3T^{2} \)
59 \( 1 - 41.3iT - 3.48e3T^{2} \)
61 \( 1 + 20.4T + 3.72e3T^{2} \)
67 \( 1 + 27.1iT - 4.48e3T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.7iT - 5.32e3T^{2} \)
79 \( 1 - 63.2T + 6.24e3T^{2} \)
83 \( 1 + 89.9T + 6.88e3T^{2} \)
89 \( 1 - 63.1iT - 7.92e3T^{2} \)
97 \( 1 - 19.1iT - 9.40e3T^{2} \)
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

−10.59674838324702636480122935214, −9.786232999456804074037865175613, −9.306385599599901091274349570808, −8.342406198486195647794591195592, −7.75826459956492602153338775815, −6.60488348089188884070645342899, −4.88828456008543954761204456845, −4.34614190532121129123610430305, −3.08157218231671675833505204568, −1.49430658549953275189948256997, 0.51384961675670253655508881327, 1.96660470912345887567370175019, 3.30896058108153887011519338116, 4.52779986446550508201064997650, 5.84131575992603927473072184239, 7.03018600041851590034532236139, 7.961863763937424386263523271877, 8.541776199069127745432346004100, 9.330823880560495332713727731586, 10.02809934037085374594927936998

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