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Properties

Label 2-1408-88.21-c2-0-57
Degree $2$
Conductor $1408$
Sign $0.636 - 0.771i$
Analytic cond. $38.3652$
Root an. cond. $6.19396$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·3-s − 23.0·9-s + (7 − 8.48i)11-s − 33.9i·17-s + 34·19-s + 25·25-s − 79.1i·27-s + (48 + 39.5i)33-s + 67.8i·41-s + 14·43-s + 49·49-s + 192·51-s + 192. i·57-s − 84.8i·59-s − 118. i·67-s + ⋯
L(s)  = 1  + 1.88i·3-s − 2.55·9-s + (0.636 − 0.771i)11-s − 1.99i·17-s + 1.78·19-s + 25-s − 2.93i·27-s + (1.45 + 1.19i)33-s + 1.65i·41-s + 0.325·43-s + 0.999·49-s + 3.76·51-s + 3.37i·57-s − 1.43i·59-s − 1.77i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(38.3652\)
Root analytic conductor: \(6.19396\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1),\ 0.636 - 0.771i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.939077292\)
\(L(\frac12)\) \(\approx\) \(1.939077292\)
\(L(2)\) 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 \( 1 + (-7 + 8.48i)T \)
good3 \( 1 - 5.65iT - 9T^{2} \)
5 \( 1 - 25T^{2} \)
7 \( 1 - 49T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 + 33.9iT - 289T^{2} \)
19 \( 1 - 34T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 67.8iT - 1.68e3T^{2} \)
43 \( 1 - 14T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + 84.8iT - 3.48e3T^{2} \)
61 \( 1 + 3.72e3T^{2} \)
67 \( 1 + 118. iT - 4.48e3T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + 33.9iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 158T + 6.88e3T^{2} \)
89 \( 1 + 146T + 7.92e3T^{2} \)
97 \( 1 - 94T + 9.40e3T^{2} \)
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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

−9.448606980672001316721585055132, −9.088734362302261376946166814132, −8.128006411941461596036553135855, −7.02093925541821268823913377361, −5.85568712001207843659091171760, −5.10039122357405202353062936362, −4.51289546072563495557478006034, −3.31688673817087959069914718301, −2.94525630093632945083133632968, −0.69600379333809195685976179718, 1.00323108978335431448540679998, 1.72259473924581420520376304892, 2.79543334521202955739796031898, 3.97977300583798891503328490699, 5.47543926195793997239775194906, 6.06779412319055621916667616111, 7.14933561271745253263486918793, 7.27309956762810144426964084154, 8.396532899344779589782947259213, 8.894351808172239973495817609974

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