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Properties

Label 2-1408-176.131-c1-0-24
Degree $2$
Conductor $1408$
Sign $0.986 - 0.166i$
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  + (0.259 − 0.259i)3-s + (1.45 − 1.45i)5-s + 0.413i·7-s + 2.86i·9-s + (−0.217 − 3.30i)11-s + (−1.40 + 1.40i)13-s − 0.754i·15-s + 7.18i·17-s + (4.93 + 4.93i)19-s + (0.107 + 0.107i)21-s + 6.32·23-s + 0.764i·25-s + (1.51 + 1.51i)27-s + (0.338 − 0.338i)29-s − 6.19i·31-s + ⋯
L(s)  = 1  + (0.149 − 0.149i)3-s + (0.650 − 0.650i)5-s + 0.156i·7-s + 0.955i·9-s + (−0.0654 − 0.997i)11-s + (−0.390 + 0.390i)13-s − 0.194i·15-s + 1.74i·17-s + (1.13 + 1.13i)19-s + (0.0233 + 0.0233i)21-s + 1.31·23-s + 0.152i·25-s + (0.292 + 0.292i)27-s + (0.0628 − 0.0628i)29-s − 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $0.986 - 0.166i$
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: \((2,\ 1408,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.030258907\)
\(L(\frac12)\) \(\approx\) \(2.030258907\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.217 + 3.30i)T \)
good3 \( 1 + (-0.259 + 0.259i)T - 3iT^{2} \)
5 \( 1 + (-1.45 + 1.45i)T - 5iT^{2} \)
7 \( 1 - 0.413iT - 7T^{2} \)
13 \( 1 + (1.40 - 1.40i)T - 13iT^{2} \)
17 \( 1 - 7.18iT - 17T^{2} \)
19 \( 1 + (-4.93 - 4.93i)T + 19iT^{2} \)
23 \( 1 - 6.32T + 23T^{2} \)
29 \( 1 + (-0.338 + 0.338i)T - 29iT^{2} \)
31 \( 1 + 6.19iT - 31T^{2} \)
37 \( 1 + (-6.64 + 6.64i)T - 37iT^{2} \)
41 \( 1 + 0.706T + 41T^{2} \)
43 \( 1 + (-2.33 + 2.33i)T - 43iT^{2} \)
47 \( 1 + 4.37iT - 47T^{2} \)
53 \( 1 + (4.01 - 4.01i)T - 53iT^{2} \)
59 \( 1 + (2.93 + 2.93i)T + 59iT^{2} \)
61 \( 1 + (6.52 - 6.52i)T - 61iT^{2} \)
67 \( 1 + (2.43 - 2.43i)T - 67iT^{2} \)
71 \( 1 + 7.09T + 71T^{2} \)
73 \( 1 - 7.56T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (7.25 + 7.25i)T + 83iT^{2} \)
89 \( 1 + 7.53iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−9.435536824737368678835686557214, −8.808084334379281808430425080006, −8.010936204597967354904271824370, −7.34149775176553730421697351203, −5.91631265793220075686444639761, −5.66714516994602025551434691205, −4.58688994385252951828194421983, −3.47305349898383849225740007114, −2.23827533326478339514735168971, −1.25352540011790107805434475936, 0.959084241010479996899321893777, 2.68774698489699425241177098026, 3.08244012782670474048200227491, 4.63599920513270008830265695561, 5.20153060120350407112596103549, 6.51581124019833747691588900850, 6.99181490016853545957271586115, 7.71323452274921280413664774313, 9.181343158255513703688282172251, 9.424681667913807749032885467352

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