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Properties

Label 2-1408-176.43-c1-0-37
Degree $2$
Conductor $1408$
Sign $0.400 + 0.916i$
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  + (0.766 + 0.766i)3-s + (0.946 + 0.946i)5-s − 0.840i·7-s − 1.82i·9-s + (0.809 − 3.21i)11-s + (−4.53 − 4.53i)13-s + 1.45i·15-s + 0.629i·17-s + (2.53 − 2.53i)19-s + (0.644 − 0.644i)21-s − 3.15·23-s − 3.20i·25-s + (3.69 − 3.69i)27-s + (−3.41 − 3.41i)29-s + 2.10i·31-s + ⋯
L(s)  = 1  + (0.442 + 0.442i)3-s + (0.423 + 0.423i)5-s − 0.317i·7-s − 0.608i·9-s + (0.244 − 0.969i)11-s + (−1.25 − 1.25i)13-s + 0.374i·15-s + 0.152i·17-s + (0.582 − 0.582i)19-s + (0.140 − 0.140i)21-s − 0.657·23-s − 0.641i·25-s + (0.711 − 0.711i)27-s + (−0.633 − 0.633i)29-s + 0.377i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.400 + 0.916i$
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: \((2,\ 1408,\ (\ :1/2),\ 0.400 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698024237\)
\(L(\frac12)\) \(\approx\) \(1.698024237\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 3.21i)T \)
good3 \( 1 + (-0.766 - 0.766i)T + 3iT^{2} \)
5 \( 1 + (-0.946 - 0.946i)T + 5iT^{2} \)
7 \( 1 + 0.840iT - 7T^{2} \)
13 \( 1 + (4.53 + 4.53i)T + 13iT^{2} \)
17 \( 1 - 0.629iT - 17T^{2} \)
19 \( 1 + (-2.53 + 2.53i)T - 19iT^{2} \)
23 \( 1 + 3.15T + 23T^{2} \)
29 \( 1 + (3.41 + 3.41i)T + 29iT^{2} \)
31 \( 1 - 2.10iT - 31T^{2} \)
37 \( 1 + (6.49 + 6.49i)T + 37iT^{2} \)
41 \( 1 + 7.84T + 41T^{2} \)
43 \( 1 + (-2.55 - 2.55i)T + 43iT^{2} \)
47 \( 1 - 3.26iT - 47T^{2} \)
53 \( 1 + (-7.27 - 7.27i)T + 53iT^{2} \)
59 \( 1 + (6.06 - 6.06i)T - 59iT^{2} \)
61 \( 1 + (-3.33 - 3.33i)T + 61iT^{2} \)
67 \( 1 + (-8.95 - 8.95i)T + 67iT^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + 3.78T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + (-9.42 + 9.42i)T - 83iT^{2} \)
89 \( 1 + 1.82iT - 89T^{2} \)
97 \( 1 + 2.21T + 97T^{2} \)
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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

−9.445182449181345144778314439164, −8.731562304724771951930568462973, −7.82101074084809742259106904647, −7.02061345438413187229391786239, −6.04834759537932756857580802003, −5.31245117627083319738260578133, −4.10666568471046997747276746059, −3.24127067101652690315890506494, −2.45355009595946558807638904727, −0.62105182015329204308635363518, 1.78204855930154677053181464814, 2.15327441759664958686422717123, 3.62707753942757530953468850886, 4.88114683763008914523733078020, 5.30391153990423649872407897114, 6.70930445881122969972741058178, 7.24558524689081187759591624309, 8.065136715230401620209957419110, 8.982342605344008813283336678574, 9.615890769856648536599994967932

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