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Properties

Label 2-1408-176.131-c1-0-29
Degree $2$
Conductor $1408$
Sign $0.650 + 0.759i$
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  + (−1.33 + 1.33i)3-s + (−1.27 + 1.27i)5-s + 0.148i·7-s − 0.586i·9-s + (1.11 − 3.12i)11-s + (−0.488 + 0.488i)13-s − 3.42i·15-s − 3.29i·17-s + (−1.88 − 1.88i)19-s + (−0.198 − 0.198i)21-s − 7.21·23-s + 1.72i·25-s + (−3.23 − 3.23i)27-s + (−4.17 + 4.17i)29-s − 2.16i·31-s + ⋯
L(s)  = 1  + (−0.773 + 0.773i)3-s + (−0.571 + 0.571i)5-s + 0.0560i·7-s − 0.195i·9-s + (0.335 − 0.942i)11-s + (−0.135 + 0.135i)13-s − 0.884i·15-s − 0.799i·17-s + (−0.432 − 0.432i)19-s + (−0.0433 − 0.0433i)21-s − 1.50·23-s + 0.345i·25-s + (−0.621 − 0.621i)27-s + (−0.774 + 0.774i)29-s − 0.388i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.650 + 0.759i$
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: \((2,\ 1408,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5929746752\)
\(L(\frac12)\) \(\approx\) \(0.5929746752\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 3.12i)T \)
good3 \( 1 + (1.33 - 1.33i)T - 3iT^{2} \)
5 \( 1 + (1.27 - 1.27i)T - 5iT^{2} \)
7 \( 1 - 0.148iT - 7T^{2} \)
13 \( 1 + (0.488 - 0.488i)T - 13iT^{2} \)
17 \( 1 + 3.29iT - 17T^{2} \)
19 \( 1 + (1.88 + 1.88i)T + 19iT^{2} \)
23 \( 1 + 7.21T + 23T^{2} \)
29 \( 1 + (4.17 - 4.17i)T - 29iT^{2} \)
31 \( 1 + 2.16iT - 31T^{2} \)
37 \( 1 + (-0.771 + 0.771i)T - 37iT^{2} \)
41 \( 1 - 8.26T + 41T^{2} \)
43 \( 1 + (-7.14 + 7.14i)T - 43iT^{2} \)
47 \( 1 + 7.43iT - 47T^{2} \)
53 \( 1 + (-6.68 + 6.68i)T - 53iT^{2} \)
59 \( 1 + (1.21 + 1.21i)T + 59iT^{2} \)
61 \( 1 + (2.33 - 2.33i)T - 61iT^{2} \)
67 \( 1 + (-7.02 + 7.02i)T - 67iT^{2} \)
71 \( 1 + 4.85T + 71T^{2} \)
73 \( 1 + 8.13T + 73T^{2} \)
79 \( 1 + 5.02T + 79T^{2} \)
83 \( 1 + (-7.65 - 7.65i)T + 83iT^{2} \)
89 \( 1 + 6.12iT - 89T^{2} \)
97 \( 1 + 6.62T + 97T^{2} \)
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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

−9.524408340651254900629175921410, −8.771962260971417631326483426560, −7.73519907149431259148351697607, −7.00973347479901744630238400196, −5.95071188060879874901495191438, −5.36728314582538512229318385737, −4.23665968508760154694470513600, −3.62472273304740303147607929355, −2.33769507121346840540932865438, −0.30958566017941180367501039504, 1.08508912088251248827116895737, 2.22063157446578657677205489547, 3.96463562157955115362351802950, 4.45864009524882799369112857079, 5.85736825306647710940710975203, 6.19269855501643991281277575519, 7.38378607881833644105034535096, 7.79940957979025408776656447252, 8.786876614876359387369678128626, 9.690163932593476401433797278350

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