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Properties

Label 2-1216-8.5-c1-0-21
Degree $2$
Conductor $1216$
Sign $0.258 + 0.965i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·3-s + 1.73i·5-s + 1.73·7-s − 9-s − 3i·11-s + 3.46·15-s − 3·17-s − i·19-s − 3.46i·21-s + 3.46·23-s + 2.00·25-s − 4i·27-s + 3.46i·29-s + 6.92·31-s − 6·33-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.774i·5-s + 0.654·7-s − 0.333·9-s − 0.904i·11-s + 0.894·15-s − 0.727·17-s − 0.229i·19-s − 0.755i·21-s + 0.722·23-s + 0.400·25-s − 0.769i·27-s + 0.643i·29-s + 1.24·31-s − 1.04·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773590533\)
\(L(\frac12)\) \(\approx\) \(1.773590533\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 5.19iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−9.432763625990370157256677661334, −8.540428649786032302996854370281, −7.83560832820863925912158877143, −7.00394066361180618803073787617, −6.50430011219516939125375699798, −5.52567955327388282498926208982, −4.36641153913946972863328227426, −3.05900881158751192129317369803, −2.14464037007539385288493675998, −0.869532506126091502286124386544, 1.34366561065017568858271144626, 2.78433399964219420287668859552, 4.26152648992979132034159970546, 4.57629126388276573532465852878, 5.30996495404645124766230089083, 6.54061604236891968259855100544, 7.58103408563278711420364678001, 8.510258927790846306545128476937, 9.134651443908583858582983202941, 9.904260175451392310280789054761

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