Properties

Label 2-1408-88.43-c1-0-11
Degree $2$
Conductor $1408$
Sign $-i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 3.31i·11-s + 2.82·13-s − 6.63i·19-s + 9.38i·23-s + 5·25-s − 8.48·29-s + 9.38i·31-s + 6.63i·43-s + 9.38i·47-s − 7·49-s + 14.1·61-s + 9.38i·71-s + 9·81-s − 6.63i·83-s + ⋯
L(s)  = 1  − 9-s + 1.00i·11-s + 0.784·13-s − 1.52i·19-s + 1.95i·23-s + 25-s − 1.57·29-s + 1.68i·31-s + 1.01i·43-s + 1.36i·47-s − 49-s + 1.81·61-s + 1.11i·71-s + 81-s − 0.728i·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-i$
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: \((2,\ 1408,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.189655330\)
\(L(\frac12)\) \(\approx\) \(1.189655330\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6.63iT - 19T^{2} \)
23 \( 1 - 9.38iT - 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 9.38iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 6.63iT - 43T^{2} \)
47 \( 1 - 9.38iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 9.38iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.487825314952556159389643836671, −9.114559200625181108969006482261, −8.182881707708371348012438552221, −7.28655426704514177034737969136, −6.57422756343468098576802943509, −5.50248388980466448592687575236, −4.86205809218635912770166615541, −3.62633993415349024223016205785, −2.73913812502171533258711311120, −1.40634542141111269094381189358, 0.49395299866851651567725660205, 2.13017551992483252965502554027, 3.28965541515578707849951865382, 4.05155508992164448297632912867, 5.42765381152160514717305868538, 5.95885773387392065252218857080, 6.76368947674859431460870959687, 8.080346007472907318376379931199, 8.422203193797042573690006350714, 9.202385075192590665756268095342

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