Properties

Label 2-1216-76.11-c0-0-1
Degree $2$
Conductor $1216$
Sign $0.305 - 0.952i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (−0.866 + 0.499i)15-s + (−0.5 + 0.866i)17-s + i·19-s + (0.866 − 0.5i)23-s i·27-s + (0.5 + 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (−0.866 + 0.499i)15-s + (−0.5 + 0.866i)17-s + i·19-s + (0.866 − 0.5i)23-s i·27-s + (0.5 + 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260453432\)
\(L(\frac12)\) \(\approx\) \(1.260453432\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.996015271613500717566228374038, −9.258738468380851951473213140809, −8.462157388716020525785505179617, −7.85390698760254343388099242505, −6.72552270229868958113104509537, −6.18204788887165868167193491315, −4.68661615658372845467334466105, −3.74725239184903231767558580199, −3.21647588991003675215054211196, −1.97461444169619673993998852549, 1.10050280817860589540909830881, 2.57476138032715204162864717431, 3.39141442100870189581158695675, 4.71071284471956799887229606276, 5.30924291287216351748220960892, 6.72814245635822106468757900617, 7.42198217272792058501705183934, 8.380998961346037792458308540163, 8.656353934983040406366919718931, 9.469381371548169058012118591293

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