↑

Properties

Label 2-1408-176.43-c1-0-25
Degree $2$
Conductor $1408$
Sign $-0.396 + 0.918i$
Analytic cond. $11.2429$
Root an. cond. $3.35304$
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  + (−0.108 − 0.108i)3-s + (−2.69 − 2.69i)5-s + 3.17i·7-s − 2.97i·9-s + (3.31 − 0.0991i)11-s + (3.01 + 3.01i)13-s + 0.582i·15-s − 3.40i·17-s + (0.948 − 0.948i)19-s + (0.342 − 0.342i)21-s − 1.13·23-s + 9.47i·25-s + (−0.646 + 0.646i)27-s + (−6.26 − 6.26i)29-s − 0.958i·31-s + ⋯
L(s)  = 1  + (−0.0624 − 0.0624i)3-s + (−1.20 − 1.20i)5-s + 1.19i·7-s − 0.992i·9-s + (0.999 − 0.0298i)11-s + (0.836 + 0.836i)13-s + 0.150i·15-s − 0.825i·17-s + (0.217 − 0.217i)19-s + (0.0748 − 0.0748i)21-s − 0.236·23-s + 1.89i·25-s + (−0.124 + 0.124i)27-s + (−1.16 − 1.16i)29-s − 0.172i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015779899\)
\(L(\frac12)\) \(\approx\) \(1.015779899\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-3.31 + 0.0991i)T \)
good3 \( 1 + (0.108 + 0.108i)T + 3iT^{2} \)
5 \( 1 + (2.69 + 2.69i)T + 5iT^{2} \)
7 \( 1 - 3.17iT - 7T^{2} \)
13 \( 1 + (-3.01 - 3.01i)T + 13iT^{2} \)
17 \( 1 + 3.40iT - 17T^{2} \)
19 \( 1 + (-0.948 + 0.948i)T - 19iT^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 + (6.26 + 6.26i)T + 29iT^{2} \)
31 \( 1 + 0.958iT - 31T^{2} \)
37 \( 1 + (-1.84 - 1.84i)T + 37iT^{2} \)
41 \( 1 + 1.95T + 41T^{2} \)
43 \( 1 + (4.09 + 4.09i)T + 43iT^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + (2.50 + 2.50i)T + 53iT^{2} \)
59 \( 1 + (-5.95 + 5.95i)T - 59iT^{2} \)
61 \( 1 + (5.84 + 5.84i)T + 61iT^{2} \)
67 \( 1 + (5.46 + 5.46i)T + 67iT^{2} \)
71 \( 1 + 8.90T + 71T^{2} \)
73 \( 1 + 4.13T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + (2.40 - 2.40i)T - 83iT^{2} \)
89 \( 1 + 5.10iT - 89T^{2} \)
97 \( 1 - 7.10T + 97T^{2} \)
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

−8.955345221934645204142910148301, −8.844605697751638260574421675079, −7.84563555712238310207491661211, −6.81270637750146535643732621465, −6.00028263054443083051637165070, −5.04616919190668358972836419936, −4.08181243566393025504341817552, −3.45782804599128620534477084366, −1.77443947852836972834675503415, −0.45768482690607312777537822774, 1.36758160964704666361489744217, 3.07718463792292664196876878065, 3.78683263535489651756340615849, 4.41382689706816309681965596753, 5.83638119987701956677641751179, 6.71892951611874295147268928893, 7.54513790306722339566428186486, 7.85211765191228788956769828446, 8.873181689843899049870103787122, 10.20145255218170491103088791604

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