Properties

Label 2-48-48.5-c2-0-12
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: $\chi_{48} (5, \cdot )$
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

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

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