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Properties

Label 2-48-48.35-c3-0-4
Degree $2$
Conductor $48$
Sign $-0.331 - 0.943i$
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  + (1.97 + 2.01i)2-s + (−0.0749 + 5.19i)3-s + (−0.159 + 7.99i)4-s + (−5.37 − 5.37i)5-s + (−10.6 + 10.1i)6-s + 14.8·7-s + (−16.4 + 15.5i)8-s + (−26.9 − 0.779i)9-s + (0.214 − 21.5i)10-s + (30.0 − 30.0i)11-s + (−41.5 − 1.42i)12-s + (61.5 + 61.5i)13-s + (29.4 + 30.0i)14-s + (28.3 − 27.5i)15-s + (−63.9 − 2.55i)16-s − 48.8i·17-s + ⋯
L(s)  = 1  + (0.700 + 0.714i)2-s + (−0.0144 + 0.999i)3-s + (−0.0199 + 0.999i)4-s + (−0.480 − 0.480i)5-s + (−0.724 + 0.689i)6-s + 0.802·7-s + (−0.727 + 0.685i)8-s + (−0.999 − 0.0288i)9-s + (0.00678 − 0.680i)10-s + (0.823 − 0.823i)11-s + (−0.999 − 0.0343i)12-s + (1.31 + 1.31i)13-s + (0.561 + 0.572i)14-s + (0.487 − 0.473i)15-s + (−0.999 − 0.0398i)16-s − 0.696i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05716 + 1.49274i\)
\(L(\frac12)\) \(\approx\) \(1.05716 + 1.49274i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 - 2.01i)T \)
3 \( 1 + (0.0749 - 5.19i)T \)
good5 \( 1 + (5.37 + 5.37i)T + 125iT^{2} \)
7 \( 1 - 14.8T + 343T^{2} \)
11 \( 1 + (-30.0 + 30.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-61.5 - 61.5i)T + 2.19e3iT^{2} \)
17 \( 1 + 48.8iT - 4.91e3T^{2} \)
19 \( 1 + (-7.45 + 7.45i)T - 6.85e3iT^{2} \)
23 \( 1 + 43.0iT - 1.21e4T^{2} \)
29 \( 1 + (32.9 - 32.9i)T - 2.43e4iT^{2} \)
31 \( 1 + 173. iT - 2.97e4T^{2} \)
37 \( 1 + (177. - 177. i)T - 5.06e4iT^{2} \)
41 \( 1 + 454.T + 6.89e4T^{2} \)
43 \( 1 + (-239. - 239. i)T + 7.95e4iT^{2} \)
47 \( 1 - 30.4T + 1.03e5T^{2} \)
53 \( 1 + (-235. - 235. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-260. + 260. i)T - 2.05e5iT^{2} \)
61 \( 1 + (388. + 388. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-334. + 334. i)T - 3.00e5iT^{2} \)
71 \( 1 - 522. iT - 3.57e5T^{2} \)
73 \( 1 + 689. iT - 3.89e5T^{2} \)
79 \( 1 - 692. iT - 4.93e5T^{2} \)
83 \( 1 + (677. + 677. i)T + 5.71e5iT^{2} \)
89 \( 1 - 261.T + 7.04e5T^{2} \)
97 \( 1 + 641.T + 9.12e5T^{2} \)
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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

−15.61912391625015752814655647342, −14.38527896178315614437193065479, −13.71928581569805714082627411300, −11.79864202368481950562544089889, −11.26520703733858027679547935609, −9.041942545589855912773264609909, −8.291908642336260248517726681917, −6.33053005227089246293128945981, −4.79116152186247802403681749170, −3.76192275748805620309235437641, 1.52081959846227868893682711386, 3.56316706647948739490182374350, 5.60015384848170258217840414563, 7.06147966286003355776454466735, 8.560116494726079166062402327021, 10.57520052452682205029813242612, 11.50487797020268619917892794764, 12.45977320187861727985456365683, 13.52775970506611714327759147345, 14.61075225892799704981723774375

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