↑

Properties

Label 2-408-17.8-c1-0-0
Degree $2$
Conductor $408$
Sign $-0.761 - 0.648i$
Analytic cond. $3.25789$
Root an. cond. $1.80496$
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  + (0.382 − 0.923i)3-s + (−3.44 − 1.42i)5-s + (−2.17 + 0.902i)7-s + (−0.707 − 0.707i)9-s + (0.746 + 1.80i)11-s + 5.01i·13-s + (−2.63 + 2.63i)15-s + (−1.56 + 3.81i)17-s + (−2.80 + 2.80i)19-s + 2.35i·21-s + (−3.53 − 8.53i)23-s + (6.28 + 6.28i)25-s + (−0.923 + 0.382i)27-s + (−6.78 − 2.80i)29-s + (1.47 − 3.56i)31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (−1.53 − 0.637i)5-s + (−0.823 + 0.340i)7-s + (−0.235 − 0.235i)9-s + (0.225 + 0.543i)11-s + 1.38i·13-s + (−0.680 + 0.680i)15-s + (−0.378 + 0.925i)17-s + (−0.642 + 0.642i)19-s + 0.514i·21-s + (−0.737 − 1.78i)23-s + (1.25 + 1.25i)25-s + (−0.177 + 0.0736i)27-s + (−1.25 − 0.521i)29-s + (0.265 − 0.640i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(408\)    =    \(2^{3} \cdot 3 \cdot 17\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(3.25789\)
Root analytic conductor: \(1.80496\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{408} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 408,\ (\ :1/2),\ -0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0411368 + 0.111798i\)
\(L(\frac12)\) \(\approx\) \(0.0411368 + 0.111798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (1.56 - 3.81i)T \)
good5 \( 1 + (3.44 + 1.42i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.17 - 0.902i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.746 - 1.80i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.01iT - 13T^{2} \)
19 \( 1 + (2.80 - 2.80i)T - 19iT^{2} \)
23 \( 1 + (3.53 + 8.53i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (6.78 + 2.80i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.47 + 3.56i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.624 - 1.50i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.00 + 2.48i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (7.41 + 7.41i)T + 43iT^{2} \)
47 \( 1 + 6.29iT - 47T^{2} \)
53 \( 1 + (7.50 - 7.50i)T - 53iT^{2} \)
59 \( 1 + (3.05 + 3.05i)T + 59iT^{2} \)
61 \( 1 + (8.06 - 3.33i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 6.10T + 67T^{2} \)
71 \( 1 + (0.839 - 2.02i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.15 - 1.72i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.55 - 11.0i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (5.67 - 5.67i)T - 83iT^{2} \)
89 \( 1 - 6.70iT - 89T^{2} \)
97 \( 1 + (1.06 + 0.442i)T + (68.5 + 68.5i)T^{2} \)
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

−11.90317602919333776670001638118, −10.87803701558303220135730613432, −9.561955693862775148111639569472, −8.670931838406107630171783598914, −8.032282972863673133887179930041, −6.94776693085099530619367682341, −6.16243286353919784834727785867, −4.39016842721206656373549807302, −3.82571505100692628888450143327, −2.06473547565380975677943236733, 0.07199047777914608200748366516, 3.16057847315065006574821202235, 3.51461299655205110753120962415, 4.81040224684590976622471426355, 6.21849468920138382317875046877, 7.37225600567332063036152263180, 7.958147426481422662652247216984, 9.111779227003179703273116153046, 10.04837431229752221225479632701, 11.07776341272652025163474819733

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