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Properties

Label 2-1408-176.131-c1-0-40
Degree $2$
Conductor $1408$
Sign $-0.165 + 0.986i$
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  + (1.82 − 1.82i)3-s + (2.57 − 2.57i)5-s + 2.30i·7-s − 3.65i·9-s + (2.04 − 2.61i)11-s + (−2.25 + 2.25i)13-s − 9.40i·15-s − 1.40i·17-s + (−5.03 − 5.03i)19-s + (4.20 + 4.20i)21-s − 3.12·23-s − 8.27i·25-s + (−1.20 − 1.20i)27-s + (−0.452 + 0.452i)29-s + 3.35i·31-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)3-s + (1.15 − 1.15i)5-s + 0.871i·7-s − 1.21i·9-s + (0.616 − 0.787i)11-s + (−0.626 + 0.626i)13-s − 2.42i·15-s − 0.340i·17-s + (−1.15 − 1.15i)19-s + (0.918 + 0.918i)21-s − 0.650·23-s − 1.65i·25-s + (−0.231 − 0.231i)27-s + (−0.0840 + 0.0840i)29-s + 0.602i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ -0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.871202538\)
\(L(\frac12)\) \(\approx\) \(2.871202538\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 2.61i)T \)
good3 \( 1 + (-1.82 + 1.82i)T - 3iT^{2} \)
5 \( 1 + (-2.57 + 2.57i)T - 5iT^{2} \)
7 \( 1 - 2.30iT - 7T^{2} \)
13 \( 1 + (2.25 - 2.25i)T - 13iT^{2} \)
17 \( 1 + 1.40iT - 17T^{2} \)
19 \( 1 + (5.03 + 5.03i)T + 19iT^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 + (0.452 - 0.452i)T - 29iT^{2} \)
31 \( 1 - 3.35iT - 31T^{2} \)
37 \( 1 + (-3.56 + 3.56i)T - 37iT^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (1.58 - 1.58i)T - 43iT^{2} \)
47 \( 1 - 7.74iT - 47T^{2} \)
53 \( 1 + (-3.56 + 3.56i)T - 53iT^{2} \)
59 \( 1 + (-2.93 - 2.93i)T + 59iT^{2} \)
61 \( 1 + (-6.88 + 6.88i)T - 61iT^{2} \)
67 \( 1 + (6.98 - 6.98i)T - 67iT^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 + 2.27T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (-1.15 - 1.15i)T + 83iT^{2} \)
89 \( 1 - 16.8iT - 89T^{2} \)
97 \( 1 - 6.48T + 97T^{2} \)
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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

−9.102747090760342369482360582868, −8.731018950638104111556728513247, −7.953047775424679966245130656636, −6.83008654377121868578542463693, −6.14916835537044884008818849011, −5.28144019053048874100160179487, −4.22319250277156755213769706771, −2.61843837278573747381896054970, −2.15798030006872964134609144746, −1.05178666444648800698532066326, 1.93819502535750134281987105243, 2.71554896495765985753291368925, 3.81778834917367370102508146097, 4.33137627260672371734804005177, 5.69900291820049743909416328749, 6.52823558110952496893039551741, 7.42701706928380594562407321660, 8.213863051389076363062859476242, 9.298774237799524700668072625136, 9.879416785162899654482990579873

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