Properties

Label 2-1216-152.37-c2-0-45
Degree $2$
Conductor $1216$
Sign $0.801 - 0.598i$
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  + 2.73·3-s + 5.26i·5-s + 4.69·7-s − 1.49·9-s − 10.2i·11-s + 17.6·13-s + 14.4i·15-s + 25.2·17-s + (−11.7 + 14.9i)19-s + 12.8·21-s + 10.4·23-s − 2.69·25-s − 28.7·27-s − 27.2·29-s − 34.2i·31-s + ⋯
L(s)  = 1  + 0.913·3-s + 1.05i·5-s + 0.670·7-s − 0.166·9-s − 0.933i·11-s + 1.35·13-s + 0.961i·15-s + 1.48·17-s + (−0.618 + 0.785i)19-s + 0.612·21-s + 0.456·23-s − 0.107·25-s − 1.06·27-s − 0.940·29-s − 1.10i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.139414193\)
\(L(\frac12)\) \(\approx\) \(3.139414193\)
\(L(2)\) 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 \( 1 + (11.7 - 14.9i)T \)
good3 \( 1 - 2.73T + 9T^{2} \)
5 \( 1 - 5.26iT - 25T^{2} \)
7 \( 1 - 4.69T + 49T^{2} \)
11 \( 1 + 10.2iT - 121T^{2} \)
13 \( 1 - 17.6T + 169T^{2} \)
17 \( 1 - 25.2T + 289T^{2} \)
23 \( 1 - 10.4T + 529T^{2} \)
29 \( 1 + 27.2T + 841T^{2} \)
31 \( 1 + 34.2iT - 961T^{2} \)
37 \( 1 - 69.6T + 1.36e3T^{2} \)
41 \( 1 - 45.1iT - 1.68e3T^{2} \)
43 \( 1 - 7.45iT - 1.84e3T^{2} \)
47 \( 1 - 45.8T + 2.20e3T^{2} \)
53 \( 1 - 18.3T + 2.80e3T^{2} \)
59 \( 1 - 71.0T + 3.48e3T^{2} \)
61 \( 1 - 75.5iT - 3.72e3T^{2} \)
67 \( 1 + 96.9T + 4.48e3T^{2} \)
71 \( 1 + 19.5iT - 5.04e3T^{2} \)
73 \( 1 + 62.7T + 5.32e3T^{2} \)
79 \( 1 - 62.4iT - 6.24e3T^{2} \)
83 \( 1 - 101. iT - 6.88e3T^{2} \)
89 \( 1 + 73.5iT - 7.92e3T^{2} \)
97 \( 1 - 107. iT - 9.40e3T^{2} \)
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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

−9.558573863682778435205313731766, −8.616413681706057200407425233644, −8.059524136002955486106788110311, −7.44546549542253455392882075421, −6.10232245245971939829145556349, −5.73577463938336823317718365561, −4.05873855168201947271124530562, −3.33687109051576198326167546057, −2.57089004103482984361735412599, −1.21540871256675617551923486780, 0.996500952720795248258743560851, 1.99438438284528047822863170033, 3.25274571496207992532291445776, 4.23455153628081322959690192185, 5.10855678086311218422974462373, 5.95625435592252756113527206104, 7.30427295395581844387079395233, 7.981562193141632780440137152332, 8.811475023753593246553723272273, 9.042505789980512690260345043284

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