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Properties

Label 2-1216-19.18-c2-0-68
Degree $2$
Conductor $1216$
Sign $0.0313 + 0.999i$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
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  + 4.27i·3-s + 7.92·5-s − 5.75·7-s − 9.24·9-s − 16.5·11-s − 7.95i·13-s + 33.8i·15-s − 18.1·17-s + (−0.595 − 18.9i)19-s − 24.5i·21-s − 7.02·23-s + 37.7·25-s − 1.04i·27-s − 30.4i·29-s − 34.9i·31-s + ⋯
L(s)  = 1  + 1.42i·3-s + 1.58·5-s − 0.822·7-s − 1.02·9-s − 1.50·11-s − 0.612i·13-s + 2.25i·15-s − 1.06·17-s + (−0.0313 − 0.999i)19-s − 1.17i·21-s − 0.305·23-s + 1.50·25-s − 0.0387i·27-s − 1.05i·29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0313 + 0.999i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ 0.0313 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4878558379\)
\(L(\frac12)\) \(\approx\) \(0.4878558379\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (0.595 + 18.9i)T \)
good3 \( 1 - 4.27iT - 9T^{2} \)
5 \( 1 - 7.92T + 25T^{2} \)
7 \( 1 + 5.75T + 49T^{2} \)
11 \( 1 + 16.5T + 121T^{2} \)
13 \( 1 + 7.95iT - 169T^{2} \)
17 \( 1 + 18.1T + 289T^{2} \)
23 \( 1 + 7.02T + 529T^{2} \)
29 \( 1 + 30.4iT - 841T^{2} \)
31 \( 1 + 34.9iT - 961T^{2} \)
37 \( 1 + 58.6iT - 1.36e3T^{2} \)
41 \( 1 - 40.7iT - 1.68e3T^{2} \)
43 \( 1 - 12.5T + 1.84e3T^{2} \)
47 \( 1 + 88.8T + 2.20e3T^{2} \)
53 \( 1 - 62.8iT - 2.80e3T^{2} \)
59 \( 1 - 51.2iT - 3.48e3T^{2} \)
61 \( 1 - 68.1T + 3.72e3T^{2} \)
67 \( 1 + 82.5iT - 4.48e3T^{2} \)
71 \( 1 + 21.9iT - 5.04e3T^{2} \)
73 \( 1 + 57.3T + 5.32e3T^{2} \)
79 \( 1 - 88.3iT - 6.24e3T^{2} \)
83 \( 1 + 0.720T + 6.88e3T^{2} \)
89 \( 1 + 30.1iT - 7.92e3T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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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

−9.608840631561352410891446777691, −8.950524514083997454196489640246, −7.83832998335442485546813208050, −6.55090930003333601263885875112, −5.77392977147223121067064806158, −5.13624037008372736126103129539, −4.25805606468919330204120649472, −2.93367473815850922799873512008, −2.32347656149775795499293220810, −0.12572352911511203882152140920, 1.53758631549176499860549347227, 2.19766329730347583752616114762, 3.13717026098116378274165312568, 4.95260063406408942093270063216, 5.77712918740720638746865832710, 6.59244881856293876530260927167, 6.92228473397464977740282323882, 8.118233444538942539650893434505, 8.805474722901334273661444815554, 9.920020641297554135633133982306

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