↑

Properties

Label 2-403-403.36-c1-0-27
Degree $2$
Conductor $403$
Sign $-0.481 + 0.876i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (−1.06 + 0.613i)2-s + 0.887·3-s + (−0.246 + 0.426i)4-s + (−0.237 + 0.137i)5-s + (−0.943 + 0.544i)6-s + (−1.34 + 0.779i)7-s − 3.06i·8-s − 2.21·9-s + (0.168 − 0.291i)10-s + (−4.65 − 2.68i)11-s + (−0.218 + 0.378i)12-s + (−0.395 − 3.58i)13-s + (0.956 − 1.65i)14-s + (−0.210 + 0.121i)15-s + (1.38 + 2.39i)16-s + (1.84 + 3.19i)17-s + ⋯
L(s)  = 1  + (−0.751 + 0.434i)2-s + 0.512·3-s + (−0.123 + 0.213i)4-s + (−0.106 + 0.0613i)5-s + (−0.385 + 0.222i)6-s + (−0.510 + 0.294i)7-s − 1.08i·8-s − 0.737·9-s + (0.0532 − 0.0922i)10-s + (−1.40 − 0.810i)11-s + (−0.0631 + 0.109i)12-s + (−0.109 − 0.993i)13-s + (0.255 − 0.442i)14-s + (−0.0544 + 0.0314i)15-s + (0.346 + 0.599i)16-s + (0.447 + 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0805456 - 0.136155i\)
\(L(\frac12)\) \(\approx\) \(0.0805456 - 0.136155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.395 + 3.58i)T \)
31 \( 1 + (-1.09 + 5.45i)T \)
good2 \( 1 + (1.06 - 0.613i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 0.887T + 3T^{2} \)
5 \( 1 + (0.237 - 0.137i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.34 - 0.779i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.65 + 2.68i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.84 - 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.56 + 0.900i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.40 + 4.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.431 + 0.747i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 - 3.05iT - 37T^{2} \)
41 \( 1 + (9.49 + 5.48i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.99 - 3.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.07iT - 47T^{2} \)
53 \( 1 + (4.52 + 7.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.03 + 1.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.168 - 0.292i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.25 + 3.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.262iT - 71T^{2} \)
73 \( 1 + (3.86 - 2.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.93 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.42 + 0.822i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.89 + 3.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.26 - 2.45i)T + (48.5 + 84.0i)T^{2} \)
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

−10.65480993903793842541836646434, −9.887946239992041653904561757303, −8.923945180084816449037166761982, −8.015538195667318930881996162736, −7.84204092643771219245188742162, −6.28985391696660338047290608748, −5.37749595714021497942332613557, −3.57005168498766077291888962911, −2.78761532778667433336274337451, −0.11591538201006005534747959825, 1.98267184408052616750742768386, 3.12398589840576698047639231493, 4.74486527075396242806563139792, 5.73747298783023340073451161075, 7.23836497827404399688536183191, 8.072250765178356908401303851030, 8.972979582751099117933684247059, 9.827347121482410510217339782353, 10.29895604366112754488449142341, 11.47176373310429759985676858608

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