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Properties

Label 2-1408-176.43-c1-0-14
Degree $2$
Conductor $1408$
Sign $0.617 - 0.786i$
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 + (3.21 − 0.809i)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 + (0.969 − 0.244i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.617 - 0.786i$
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: \((2,\ 1408,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322537741\)
\(L(\frac12)\) \(\approx\) \(2.322537741\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.21 + 0.809i)T \)
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

−9.522270411905132704421357203773, −8.852794520797308699116468500942, −8.488888219953976782953883035416, −7.02113060047482106219773946221, −6.36900801267388369172921907653, −5.77711210539184889459354405140, −4.18109190540855522635798980928, −3.82818159503117215493247082562, −2.60708456329974397727530088243, −1.40018039970745938902057768489, 1.04981597652888043274326206435, 2.05253497220320823135797464069, 3.28673882163420417328109975698, 4.28080960156611074609859326559, 5.33162089828842099604412821519, 6.19929977695530194610170458720, 7.01784382905351262904781822621, 8.044828116714548967870016613843, 8.479615103897806967079451573335, 9.324657549097816267640181224038

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