Properties

Label 2-1408-88.43-c1-0-27
Degree $2$
Conductor $1408$
Sign $0.904 - 0.426i$
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  + 2.82·3-s + 5.00·9-s + (1.41 + 3i)11-s + 5.65i·17-s + 2i·19-s + 5·25-s + 5.65·27-s + (4.00 + 8.48i)33-s − 11.3i·41-s − 10i·43-s − 7·49-s + 16.0i·51-s + 5.65i·57-s + 14.1·59-s + 8.48·67-s + ⋯
L(s)  = 1  + 1.63·3-s + 1.66·9-s + (0.426 + 0.904i)11-s + 1.37i·17-s + 0.458i·19-s + 25-s + 1.08·27-s + (0.696 + 1.47i)33-s − 1.76i·41-s − 1.52i·43-s − 49-s + 2.24i·51-s + 0.749i·57-s + 1.84·59-s + 1.03·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.085023308\)
\(L(\frac12)\) \(\approx\) \(3.085023308\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 3i)T \)
good3 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−9.497044712354676300880217338472, −8.677641622212110419469288868989, −8.239391510131021877967420183364, −7.29909506178195020559173869761, −6.65558122589162475371907278853, −5.35399014448285097058116714963, −4.11365017378342522745373497110, −3.61707659447007285347013467205, −2.41124701856803431607961119967, −1.60936960349734452831270489327, 1.17478332145493555710386718957, 2.65797077235120219091571666030, 3.11698561832902122456914914289, 4.18297576374913806686871189126, 5.14716656469764117334284288157, 6.48280702931901403855735952025, 7.20782663919586974783954162691, 8.143400681390915179848134247238, 8.618675328743796192523585686972, 9.437128415267105060555376660250

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