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Properties

Label 2-1408-176.43-c1-0-11
Degree $2$
Conductor $1408$
Sign $0.427 - 0.904i$
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  + (1.64 + 1.64i)3-s + (−1.97 − 1.97i)5-s − 1.30i·7-s + 2.39i·9-s + (−1.98 + 2.65i)11-s + (3.45 + 3.45i)13-s − 6.47i·15-s + 6.16i·17-s + (4.15 − 4.15i)19-s + (2.14 − 2.14i)21-s − 2.00·23-s + 2.76i·25-s + (0.994 − 0.994i)27-s + (4.66 + 4.66i)29-s + 6.42i·31-s + ⋯
L(s)  = 1  + (0.948 + 0.948i)3-s + (−0.881 − 0.881i)5-s − 0.493i·7-s + 0.798i·9-s + (−0.597 + 0.801i)11-s + (0.957 + 0.957i)13-s − 1.67i·15-s + 1.49i·17-s + (0.952 − 0.952i)19-s + (0.467 − 0.467i)21-s − 0.418·23-s + 0.553i·25-s + (0.191 − 0.191i)27-s + (0.866 + 0.866i)29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.427 - 0.904i$
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: \((2,\ 1408,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945293434\)
\(L(\frac12)\) \(\approx\) \(1.945293434\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 - 2.65i)T \)
good3 \( 1 + (-1.64 - 1.64i)T + 3iT^{2} \)
5 \( 1 + (1.97 + 1.97i)T + 5iT^{2} \)
7 \( 1 + 1.30iT - 7T^{2} \)
13 \( 1 + (-3.45 - 3.45i)T + 13iT^{2} \)
17 \( 1 - 6.16iT - 17T^{2} \)
19 \( 1 + (-4.15 + 4.15i)T - 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (-4.66 - 4.66i)T + 29iT^{2} \)
31 \( 1 - 6.42iT - 31T^{2} \)
37 \( 1 + (-1.08 - 1.08i)T + 37iT^{2} \)
41 \( 1 - 4.81T + 41T^{2} \)
43 \( 1 + (-5.77 - 5.77i)T + 43iT^{2} \)
47 \( 1 + 4.74iT - 47T^{2} \)
53 \( 1 + (-3.23 - 3.23i)T + 53iT^{2} \)
59 \( 1 + (1.19 - 1.19i)T - 59iT^{2} \)
61 \( 1 + (1.81 + 1.81i)T + 61iT^{2} \)
67 \( 1 + (1.51 + 1.51i)T + 67iT^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 5.86T + 73T^{2} \)
79 \( 1 - 8.60T + 79T^{2} \)
83 \( 1 + (6.28 - 6.28i)T - 83iT^{2} \)
89 \( 1 - 6.63iT - 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−9.526863918212978995972028362540, −8.826438766387361524850094945081, −8.342900126518625923498039880331, −7.54768336443904276108137204652, −6.55771849506843641648530457825, −5.16185729119747788780984290332, −4.26259467901680157227798517584, −3.98673068704708084907065973708, −2.85821109865390498670331662368, −1.30377153473871995551242675924, 0.802722642887617979388635440126, 2.53039290333033211190970299546, 3.00404433355579890574364276731, 3.88406720948461946437251894517, 5.47568480451771601047662829348, 6.21100562287723291630302111227, 7.41079234043761617855963161986, 7.71725522206142815265043661252, 8.321719082346210070178343094790, 9.180408763213889995639380119590

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