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Properties

Label 2-48-12.11-c3-0-0
Degree $2$
Conductor $48$
Sign $-0.182 - 0.983i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 − 1.73i)3-s + 16.9i·5-s + 17.3i·7-s + (20.9 + 16.9i)9-s − 29.3·11-s − 26·13-s + (29.3 − 83.1i)15-s − 67.8i·17-s + 107. i·19-s + (30 − 84.8i)21-s + 176.·23-s − 162.·25-s + (−73.4 − 119. i)27-s − 16.9i·29-s + 31.1i·31-s + ⋯
L(s)  = 1  + (−0.942 − 0.333i)3-s + 1.51i·5-s + 0.935i·7-s + (0.777 + 0.628i)9-s − 0.805·11-s − 0.554·13-s + (0.505 − 1.43i)15-s − 0.968i·17-s + 1.29i·19-s + (0.311 − 0.881i)21-s + 1.59·23-s − 1.30·25-s + (−0.523 − 0.851i)27-s − 0.108i·29-s + 0.180i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.182 - 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.519043 + 0.624400i\)
\(L(\frac12)\) \(\approx\) \(0.519043 + 0.624400i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.89 + 1.73i)T \)
good5 \( 1 - 16.9iT - 125T^{2} \)
7 \( 1 - 17.3iT - 343T^{2} \)
11 \( 1 + 29.3T + 1.33e3T^{2} \)
13 \( 1 + 26T + 2.19e3T^{2} \)
17 \( 1 + 67.8iT - 4.91e3T^{2} \)
19 \( 1 - 107. iT - 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 + 16.9iT - 2.43e4T^{2} \)
31 \( 1 - 31.1iT - 2.97e4T^{2} \)
37 \( 1 - 206T + 5.06e4T^{2} \)
41 \( 1 - 305. iT - 6.89e4T^{2} \)
43 \( 1 - 93.5iT - 7.95e4T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 + 50.9iT - 1.48e5T^{2} \)
59 \( 1 - 558.T + 2.05e5T^{2} \)
61 \( 1 - 278T + 2.26e5T^{2} \)
67 \( 1 + 890. iT - 3.00e5T^{2} \)
71 \( 1 - 58.7T + 3.57e5T^{2} \)
73 \( 1 + 422T + 3.89e5T^{2} \)
79 \( 1 - 668. iT - 4.93e5T^{2} \)
83 \( 1 - 29.3T + 5.71e5T^{2} \)
89 \( 1 - 373. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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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

−15.42692823388726485147027448659, −14.50558179897086988247988525037, −13.03113880415047416560639121829, −11.82279775493110361399822747953, −10.92304465276536632813241087427, −9.832731730861270285636002974219, −7.69356401754218641905827460137, −6.56625715956571960629456174105, −5.28333118641000321052146491692, −2.67978124865639315022734160496, 0.71455361788510868503819984830, 4.39581365844079065872555497017, 5.35544127540697472462002138247, 7.19896939401082726898044586355, 8.848649501376301939389858614993, 10.16118561804820831902803443080, 11.28145290676356599116562575691, 12.69534073836030898833525085200, 13.22818853387690027585601505297, 15.16685859460070396103439266991

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