↑

Properties

Label 2-525-35.9-c1-0-20
Degree $2$
Conductor $525$
Sign $0.619 - 0.784i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.11 + 1.21i)2-s + (0.866 − 0.5i)3-s + (1.97 + 3.41i)4-s + 2.43·6-s + (2.46 − 0.970i)7-s + 4.73i·8-s + (0.499 − 0.866i)9-s + (−2.29 − 3.96i)11-s + (3.41 + 1.97i)12-s + 1.35i·13-s + (6.37 + 0.951i)14-s + (−1.82 + 3.15i)16-s + (−3.71 + 2.14i)17-s + (2.11 − 1.21i)18-s + (−3.40 + 5.90i)19-s + ⋯
L(s)  = 1  + (1.49 + 0.861i)2-s + (0.499 − 0.288i)3-s + (0.985 + 1.70i)4-s + 0.995·6-s + (0.930 − 0.366i)7-s + 1.67i·8-s + (0.166 − 0.288i)9-s + (−0.690 − 1.19i)11-s + (0.985 + 0.568i)12-s + 0.376i·13-s + (1.70 + 0.254i)14-s + (−0.455 + 0.789i)16-s + (−0.901 + 0.520i)17-s + (0.497 − 0.287i)18-s + (−0.781 + 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.619 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.38878 + 1.64229i\)
\(L(\frac12)\) \(\approx\) \(3.38878 + 1.64229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.46 + 0.970i)T \)
good2 \( 1 + (-2.11 - 1.21i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (2.29 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (3.71 - 2.14i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.40 - 5.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.36 + 1.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-4.05 - 7.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.1 + 5.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 - 3.52iT - 43T^{2} \)
47 \( 1 + (2.85 + 1.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.18 + 4.72i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.350 - 0.606i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.68 - 0.970i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (5.33 - 3.08i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.08 - 5.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + (-0.146 + 0.253i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.00iT - 97T^{2} \)
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

−11.20553363973611573013659110309, −10.37609394791926969820919494445, −8.580480470705733104809985077073, −8.170016509480245550960016203630, −7.19879291870860536478628037474, −6.25652804212210409516900023534, −5.42611555178805849346356666017, −4.30602843136764032115689845567, −3.54541481948796887469954855429, −2.10571859529667228071631049734, 2.05230848909953242746122392930, 2.63289590937365693647812815370, 4.15308724011159349068800006759, 4.74846758315483397312893147386, 5.55135190904788816551113698577, 6.92665888533194254282646694499, 8.028299057511340862251293572117, 9.166576377724901493794540136675, 10.20771285222048416960024462705, 10.96073990811673441196197099743

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