Properties

Label 2-1408-88.43-c1-0-30
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

\(L(1)\) \(\approx\) \(1.031782597\)
\(L(\frac12)\) \(\approx\) \(1.031782597\)
\(L(\frac{3}{2})\) 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

−9.289503377275354766456258255071, −8.618240857648820936938929844593, −7.77671910715295337164440342028, −6.82518533542599136092511956491, −6.22783085235467156227179405390, −4.84747800231023645809864004616, −4.60130616550881524474584273315, −2.93088471236026239052555828914, −2.33216948850350465641759074140, −0.42918242374921526987619821287, 1.33788788936343545814428595684, 2.88063971701123431364709843756, 3.48801615842988575536236348281, 4.87598968426192542203092021489, 5.62571351999058326321695828157, 6.36921521521164189141864332501, 7.42455432744198969041345732111, 8.250841851099269630160513582560, 8.834390401168611827234273154960, 9.753449856836599360909958940716

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