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Properties

Label 2-1408-176.43-c1-0-12
Degree $2$
Conductor $1408$
Sign $-0.528 - 0.849i$
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  + (2.13 + 2.13i)3-s + (−0.987 − 0.987i)5-s + 3.85i·7-s + 6.09i·9-s + (3.00 − 1.39i)11-s + (−0.361 − 0.361i)13-s − 4.21i·15-s + 6.94i·17-s + (0.580 − 0.580i)19-s + (−8.21 + 8.21i)21-s + 1.27·23-s − 3.05i·25-s + (−6.60 + 6.60i)27-s + (−2.37 − 2.37i)29-s − 3.48i·31-s + ⋯
L(s)  = 1  + (1.23 + 1.23i)3-s + (−0.441 − 0.441i)5-s + 1.45i·7-s + 2.03i·9-s + (0.907 − 0.420i)11-s + (−0.100 − 0.100i)13-s − 1.08i·15-s + 1.68i·17-s + (0.133 − 0.133i)19-s + (−1.79 + 1.79i)21-s + 0.265·23-s − 0.610i·25-s + (−1.27 + 1.27i)27-s + (−0.440 − 0.440i)29-s − 0.625i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.528 - 0.849i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ -0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326467653\)
\(L(\frac12)\) \(\approx\) \(2.326467653\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.00 + 1.39i)T \)
good3 \( 1 + (-2.13 - 2.13i)T + 3iT^{2} \)
5 \( 1 + (0.987 + 0.987i)T + 5iT^{2} \)
7 \( 1 - 3.85iT - 7T^{2} \)
13 \( 1 + (0.361 + 0.361i)T + 13iT^{2} \)
17 \( 1 - 6.94iT - 17T^{2} \)
19 \( 1 + (-0.580 + 0.580i)T - 19iT^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 + (2.37 + 2.37i)T + 29iT^{2} \)
31 \( 1 + 3.48iT - 31T^{2} \)
37 \( 1 + (-3.84 - 3.84i)T + 37iT^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 + (3.18 + 3.18i)T + 43iT^{2} \)
47 \( 1 - 9.03iT - 47T^{2} \)
53 \( 1 + (4.45 + 4.45i)T + 53iT^{2} \)
59 \( 1 + (4.40 - 4.40i)T - 59iT^{2} \)
61 \( 1 + (-9.97 - 9.97i)T + 61iT^{2} \)
67 \( 1 + (2.18 + 2.18i)T + 67iT^{2} \)
71 \( 1 - 4.81T + 71T^{2} \)
73 \( 1 - 8.04T + 73T^{2} \)
79 \( 1 + 0.610T + 79T^{2} \)
83 \( 1 + (-5.05 + 5.05i)T - 83iT^{2} \)
89 \( 1 - 1.45iT - 89T^{2} \)
97 \( 1 - 7.42T + 97T^{2} \)
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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

−9.576328970283167150308023896732, −8.976039723152746832818856764750, −8.388669384888342003724460249119, −8.008505833729979077149641117100, −6.41423925151513308472041069243, −5.52710372046257204126321372724, −4.52323289577530755962163575766, −3.81315251001146336928336205598, −2.96060097237361543065575741871, −1.89893248268356741126014324482, 0.831652843104926063449440319043, 1.90585762403485115879321437146, 3.21124908945174736301434911293, 3.70809727228139169191171990670, 4.91676873496947933299362904820, 6.66076721887297689156504026686, 7.03600475707927047174807571028, 7.46635864028585012896594761958, 8.271816049973977307108677393237, 9.258728809420529638876755197945

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