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Properties

Label 2-1216-152.37-c0-0-0
Degree $2$
Conductor $1216$
Sign $-0.965 - 0.258i$
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  + 1.73i·5-s − 1.73·7-s − 9-s − i·11-s − 17-s + i·19-s − 1.99·25-s − 2.99i·35-s + i·43-s − 1.73i·45-s − 1.73·47-s + 1.99·49-s + 1.73·55-s + 1.73i·61-s + 1.73·63-s + ⋯
L(s)  = 1  + 1.73i·5-s − 1.73·7-s − 9-s − i·11-s − 17-s + i·19-s − 1.99·25-s − 2.99i·35-s + i·43-s − 1.73i·45-s − 1.73·47-s + 1.99·49-s + 1.73·55-s + 1.73i·61-s + 1.73·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ -0.965 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3681415366\)
\(L(\frac12)\) \(\approx\) \(0.3681415366\)
\(L(1)\) 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 \( 1 + T^{2} \)
5 \( 1 - 1.73iT - T^{2} \)
7 \( 1 + 1.73T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + 1.73T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.29317455938830649603549838509, −9.654455534655563961715526074284, −8.730815576743668873628068543434, −7.78187276211122550510255714208, −6.64934749368689905267604189069, −6.37512050132226794038680372742, −5.65278155660643339239987732125, −3.80143317610510809605380709422, −3.15437356169946264646723041268, −2.54668252330011589842346845765, 0.29217866896677768525668913622, 2.14279730663332803716379781987, 3.36803280152745760360360494033, 4.50102483581328217115477055380, 5.21404517380726172785640661391, 6.21420698267797631279798655882, 6.95133679846000348904702588559, 8.142900222549718188744348110018, 9.006078196330317939498643754640, 9.307509489928913914111322053629

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