↑

Properties

Label 2-525-21.5-c1-0-15
Degree $2$
Conductor $525$
Sign $-0.571 - 0.820i$
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  + (2.36 + 1.36i)2-s + (−1.63 − 0.583i)3-s + (2.73 + 4.74i)4-s + (−3.06 − 3.61i)6-s + (−2.24 + 1.39i)7-s + 9.49i·8-s + (2.31 + 1.90i)9-s + (1.79 − 1.03i)11-s + (−1.69 − 9.32i)12-s + 2.04i·13-s + (−7.22 + 0.226i)14-s + (−7.50 + 13.0i)16-s + (1.05 + 1.82i)17-s + (2.88 + 7.67i)18-s + (−2.41 − 1.39i)19-s + ⋯
L(s)  = 1  + (1.67 + 0.966i)2-s + (−0.941 − 0.336i)3-s + (1.36 + 2.37i)4-s + (−1.25 − 1.47i)6-s + (−0.849 + 0.526i)7-s + 3.35i·8-s + (0.773 + 0.634i)9-s + (0.540 − 0.312i)11-s + (−0.490 − 2.69i)12-s + 0.566i·13-s + (−1.93 + 0.0604i)14-s + (−1.87 + 3.25i)16-s + (0.255 + 0.441i)17-s + (0.681 + 1.80i)18-s + (−0.553 − 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17406 + 2.24791i\)
\(L(\frac12)\) \(\approx\) \(1.17406 + 2.24791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 + 0.583i)T \)
5 \( 1 \)
7 \( 1 + (2.24 - 1.39i)T \)
good2 \( 1 + (-2.36 - 1.36i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.79 + 1.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.04iT - 13T^{2} \)
17 \( 1 + (-1.05 - 1.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.53 + 0.888i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (-6.04 + 3.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.03 + 5.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 - 4.37T + 43T^{2} \)
47 \( 1 + (1.47 - 2.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.36 + 3.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.44 + 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.30 + 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.70 + 6.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.80iT - 71T^{2} \)
73 \( 1 + (5.81 - 3.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.96T + 83T^{2} \)
89 \( 1 + (6.20 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.97iT - 97T^{2} \)
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

−11.60637784233841676316120825454, −10.76336887917966490825051442865, −9.229697255303542112358631538527, −7.995379245793638940466319886270, −7.03318719432697835067800205522, −6.18735308851797495139709012626, −5.92718114856331112142511518044, −4.68831935075556822922602291154, −3.84395028483076216244838760794, −2.43566567677352751772516644496, 1.03819961434114609252795123804, 2.83537618698311410483779257293, 3.96412568189273028074469644135, 4.57207082547199318531319425377, 5.77084316803021444417742011702, 6.31619752401275045547886617785, 7.26098365567099400141209515482, 9.471555449855786061368593574650, 10.16449638199841995792157807848, 10.70504754265835995199253081199

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