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Properties

Label 2-48-48.29-c2-0-6
Degree $2$
Conductor $48$
Sign $0.513 - 0.858i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
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  + (1.21 + 1.59i)2-s + (2.77 + 1.14i)3-s + (−1.06 + 3.85i)4-s + (−4.80 − 4.80i)5-s + (1.52 + 5.80i)6-s − 7.36i·7-s + (−7.42 + 2.97i)8-s + (6.35 + 6.37i)9-s + (1.82 − 13.4i)10-s + (0.514 + 0.514i)11-s + (−7.38 + 9.46i)12-s + (7.12 + 7.12i)13-s + (11.7 − 8.91i)14-s + (−7.79 − 18.8i)15-s + (−13.7 − 8.21i)16-s + 11.1i·17-s + ⋯
L(s)  = 1  + (0.605 + 0.795i)2-s + (0.923 + 0.383i)3-s + (−0.266 + 0.963i)4-s + (−0.960 − 0.960i)5-s + (0.254 + 0.967i)6-s − 1.05i·7-s + (−0.928 + 0.372i)8-s + (0.706 + 0.707i)9-s + (0.182 − 1.34i)10-s + (0.0467 + 0.0467i)11-s + (−0.615 + 0.788i)12-s + (0.548 + 0.548i)13-s + (0.836 − 0.637i)14-s + (−0.519 − 1.25i)15-s + (−0.858 − 0.513i)16-s + 0.653i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.513 - 0.858i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.513 - 0.858i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40107 + 0.794198i\)
\(L(\frac12)\) \(\approx\) \(1.40107 + 0.794198i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 1.59i)T \)
3 \( 1 + (-2.77 - 1.14i)T \)
good5 \( 1 + (4.80 + 4.80i)T + 25iT^{2} \)
7 \( 1 + 7.36iT - 49T^{2} \)
11 \( 1 + (-0.514 - 0.514i)T + 121iT^{2} \)
13 \( 1 + (-7.12 - 7.12i)T + 169iT^{2} \)
17 \( 1 - 11.1iT - 289T^{2} \)
19 \( 1 + (21.1 + 21.1i)T + 361iT^{2} \)
23 \( 1 - 7.80T + 529T^{2} \)
29 \( 1 + (34.6 - 34.6i)T - 841iT^{2} \)
31 \( 1 - 24.8T + 961T^{2} \)
37 \( 1 + (18.2 - 18.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 64.2T + 1.68e3T^{2} \)
43 \( 1 + (-7.24 + 7.24i)T - 1.84e3iT^{2} \)
47 \( 1 + 23.0iT - 2.20e3T^{2} \)
53 \( 1 + (31.9 + 31.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (17.6 + 17.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (12.3 + 12.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-41.1 - 41.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 25.6T + 5.04e3T^{2} \)
73 \( 1 - 56.1iT - 5.32e3T^{2} \)
79 \( 1 + 35.7T + 6.24e3T^{2} \)
83 \( 1 + (-94.9 + 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 44.8T + 7.92e3T^{2} \)
97 \( 1 + 82.3T + 9.40e3T^{2} \)
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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

−15.59062826710203598096416209923, −14.59557776258011508980791358438, −13.43994092017919982631315908396, −12.67946623563444258561797849984, −11.01309957781661628756671400934, −9.037684771930685727887982380944, −8.192348944495722085720625780583, −7.01830998929766648846816719252, −4.62941201002496917518600743374, −3.81081678404030566707089731590, 2.57290149009258222400805814303, 3.84558422106880570839538960149, 6.17265009401405644474985822327, 7.88014281253038771007392890148, 9.251765794943370993562765775490, 10.75710014478680419026466551198, 11.90839520452777616350138458851, 12.84261132513800203458311907141, 14.15750795512629055034529870105, 15.06812276867586793164888277708

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