Properties

Label 2-1216-19.18-c2-0-30
Degree $2$
Conductor $1216$
Sign $0.386 - 0.922i$
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  + 0.512i·3-s + 2.30·5-s + 4.59·7-s + 8.73·9-s − 13.7·11-s + 22.2i·13-s + 1.18i·15-s + 5.43·17-s + (7.35 − 17.5i)19-s + 2.35i·21-s + 24.9·23-s − 19.6·25-s + 9.08i·27-s + 34.7i·29-s + 41.2i·31-s + ⋯
L(s)  = 1  + 0.170i·3-s + 0.461·5-s + 0.656·7-s + 0.970·9-s − 1.25·11-s + 1.71i·13-s + 0.0787i·15-s + 0.319·17-s + (0.386 − 0.922i)19-s + 0.112i·21-s + 1.08·23-s − 0.787·25-s + 0.336i·27-s + 1.19i·29-s + 1.33i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.386 - 0.922i$
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: \((2,\ 1216,\ (\ :1),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.225724068\)
\(L(\frac12)\) \(\approx\) \(2.225724068\)
\(L(2)\) 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 \( 1 + (-7.35 + 17.5i)T \)
good3 \( 1 - 0.512iT - 9T^{2} \)
5 \( 1 - 2.30T + 25T^{2} \)
7 \( 1 - 4.59T + 49T^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 - 22.2iT - 169T^{2} \)
17 \( 1 - 5.43T + 289T^{2} \)
23 \( 1 - 24.9T + 529T^{2} \)
29 \( 1 - 34.7iT - 841T^{2} \)
31 \( 1 - 41.2iT - 961T^{2} \)
37 \( 1 + 20.5iT - 1.36e3T^{2} \)
41 \( 1 - 27.8iT - 1.68e3T^{2} \)
43 \( 1 - 10.8T + 1.84e3T^{2} \)
47 \( 1 + 33.8T + 2.20e3T^{2} \)
53 \( 1 - 29.2iT - 2.80e3T^{2} \)
59 \( 1 + 65.8iT - 3.48e3T^{2} \)
61 \( 1 - 107.T + 3.72e3T^{2} \)
67 \( 1 + 1.78iT - 4.48e3T^{2} \)
71 \( 1 + 95.8iT - 5.04e3T^{2} \)
73 \( 1 - 65.4T + 5.32e3T^{2} \)
79 \( 1 + 37.2iT - 6.24e3T^{2} \)
83 \( 1 - 32.6T + 6.88e3T^{2} \)
89 \( 1 - 74.3iT - 7.92e3T^{2} \)
97 \( 1 - 128. iT - 9.40e3T^{2} \)
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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

−9.574381608409502054370725321559, −9.085358048994838256731178301119, −8.019284417228604679211465048372, −7.16575223167420959485667753999, −6.54388731349975543459494508099, −5.01425513129998946904959108526, −4.92739880561666323783916106582, −3.56936884621608761712814779970, −2.27232995297995693654506991583, −1.31690430358357537681731919151, 0.69633376871348169151216139132, 1.94986336236992861028353851071, 3.00858838901977980644673901411, 4.23781057468996937675343467005, 5.37373746929457797073625693917, 5.73446877878606576994534871442, 7.08152400909981276779368327716, 7.981798402053730480130431440985, 8.134362337209662298952031940973, 9.781142173296095881942946289739

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