Properties

Label 2-1408-88.43-c1-0-29
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

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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

\(L(1)\) \(\approx\) \(0.2192458560\)
\(L(\frac12)\) \(\approx\) \(0.2192458560\)
\(L(\frac{3}{2})\) 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 \( 1 + 10T + 97T^{2} \)
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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

−9.284447122556799742719594328892, −8.382203744558410826566439119393, −7.44028613189273682705753217738, −6.42582662587775453550679157224, −5.98522898270704782586369749591, −5.14252769636614615656199984058, −4.34969761912690698626177881418, −3.11472986555328318776700767475, −1.44637648437853610515470390283, −0.12392439755203290766652259253, 1.30036754980955717982717452346, 2.81018047411456554670468211702, 4.34268895522698667531425595810, 5.00085047111000593626639789701, 5.64846287837825106954886516346, 6.68639453885800202982140432643, 7.14771240274466937476334726444, 8.146021694604307440831733082071, 9.418297088404837919639259420268, 9.998890838575228152867133163291

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