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Properties

Label 2-1408-176.131-c1-0-4
Degree $2$
Conductor $1408$
Sign $-0.991 - 0.126i$
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  + (−2.21 + 2.21i)3-s + (0.858 − 0.858i)5-s − 1.49i·7-s − 6.85i·9-s + (−1.33 + 3.03i)11-s + (2.98 − 2.98i)13-s + 3.80i·15-s + 2.42i·17-s + (−3.38 − 3.38i)19-s + (3.31 + 3.31i)21-s + 0.410·23-s + 3.52i·25-s + (8.55 + 8.55i)27-s + (−6.31 + 6.31i)29-s + 8.16i·31-s + ⋯
L(s)  = 1  + (−1.28 + 1.28i)3-s + (0.383 − 0.383i)5-s − 0.564i·7-s − 2.28i·9-s + (−0.401 + 0.915i)11-s + (0.827 − 0.827i)13-s + 0.983i·15-s + 0.588i·17-s + (−0.775 − 0.775i)19-s + (0.723 + 0.723i)21-s + 0.0855·23-s + 0.705i·25-s + (1.64 + 1.64i)27-s + (−1.17 + 1.17i)29-s + 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 1408,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4406576053\)
\(L(\frac12)\) \(\approx\) \(0.4406576053\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 3.03i)T \)
good3 \( 1 + (2.21 - 2.21i)T - 3iT^{2} \)
5 \( 1 + (-0.858 + 0.858i)T - 5iT^{2} \)
7 \( 1 + 1.49iT - 7T^{2} \)
13 \( 1 + (-2.98 + 2.98i)T - 13iT^{2} \)
17 \( 1 - 2.42iT - 17T^{2} \)
19 \( 1 + (3.38 + 3.38i)T + 19iT^{2} \)
23 \( 1 - 0.410T + 23T^{2} \)
29 \( 1 + (6.31 - 6.31i)T - 29iT^{2} \)
31 \( 1 - 8.16iT - 31T^{2} \)
37 \( 1 + (1.37 - 1.37i)T - 37iT^{2} \)
41 \( 1 + 4.70T + 41T^{2} \)
43 \( 1 + (-2.16 + 2.16i)T - 43iT^{2} \)
47 \( 1 - 0.963iT - 47T^{2} \)
53 \( 1 + (4.60 - 4.60i)T - 53iT^{2} \)
59 \( 1 + (2.45 + 2.45i)T + 59iT^{2} \)
61 \( 1 + (-6.00 + 6.00i)T - 61iT^{2} \)
67 \( 1 + (5.95 - 5.95i)T - 67iT^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (-1.12 - 1.12i)T + 83iT^{2} \)
89 \( 1 - 5.40iT - 89T^{2} \)
97 \( 1 + 5.47T + 97T^{2} \)
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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

−10.22791725556674390956984630692, −9.282297196342752229220764494765, −8.664996604205334507331860397898, −7.31875574983025204475695136378, −6.48024900758234952099498070624, −5.53127729295276437277481148228, −5.03298815006212380590330639625, −4.19400377417884892616458190059, −3.30960251960878573061191917138, −1.37182186048846842549594857504, 0.21899381546616242411862612019, 1.68548037822081019256810572937, 2.54608535332290513891637898127, 4.14442901754677724397783299626, 5.44843649444659215626056120188, 6.05988621226139142772541896960, 6.38841589271056156637809841669, 7.44050971680765104795551405528, 8.156979320466390652844233931550, 9.072931370100397346608235778263

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