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Properties

Label 2-48-48.11-c1-0-3
Degree $2$
Conductor $48$
Sign $0.852 + 0.522i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
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  + (0.607 − 1.27i)2-s + (0.0835 + 1.73i)3-s + (−1.26 − 1.55i)4-s + (0.431 − 0.431i)5-s + (2.26 + 0.944i)6-s − 3.10·7-s + (−2.74 + 0.669i)8-s + (−2.98 + 0.289i)9-s + (−0.289 − 0.813i)10-s + (2.98 + 2.98i)11-s + (2.57 − 2.31i)12-s + (2.10 − 2.10i)13-s + (−1.88 + 3.96i)14-s + (0.782 + 0.710i)15-s + (−0.813 + 3.91i)16-s − 2.42i·17-s + ⋯
L(s)  = 1  + (0.429 − 0.903i)2-s + (0.0482 + 0.998i)3-s + (−0.631 − 0.775i)4-s + (0.193 − 0.193i)5-s + (0.922 + 0.385i)6-s − 1.17·7-s + (−0.971 + 0.236i)8-s + (−0.995 + 0.0963i)9-s + (−0.0914 − 0.257i)10-s + (0.900 + 0.900i)11-s + (0.744 − 0.667i)12-s + (0.583 − 0.583i)13-s + (−0.503 + 1.05i)14-s + (0.202 + 0.183i)15-s + (−0.203 + 0.979i)16-s − 0.589i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870465 - 0.245253i\)
\(L(\frac12)\) \(\approx\) \(0.870465 - 0.245253i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.607 + 1.27i)T \)
3 \( 1 + (-0.0835 - 1.73i)T \)
good5 \( 1 + (-0.431 + 0.431i)T - 5iT^{2} \)
7 \( 1 + 3.10T + 7T^{2} \)
11 \( 1 + (-2.98 - 2.98i)T + 11iT^{2} \)
13 \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \)
17 \( 1 + 2.42iT - 17T^{2} \)
19 \( 1 + (0.710 + 0.710i)T + 19iT^{2} \)
23 \( 1 + 5.97iT - 23T^{2} \)
29 \( 1 + (-2.86 - 2.86i)T + 29iT^{2} \)
31 \( 1 + 0.524iT - 31T^{2} \)
37 \( 1 + (-1.52 - 1.52i)T + 37iT^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 + (-0.710 + 0.710i)T - 43iT^{2} \)
47 \( 1 + 7.53T + 47T^{2} \)
53 \( 1 + (8.83 - 8.83i)T - 53iT^{2} \)
59 \( 1 + (0.0804 + 0.0804i)T + 59iT^{2} \)
61 \( 1 + (5.72 - 5.72i)T - 61iT^{2} \)
67 \( 1 + (0.391 + 0.391i)T + 67iT^{2} \)
71 \( 1 - 5.01iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 3.47iT - 79T^{2} \)
83 \( 1 + (-4.55 + 4.55i)T - 83iT^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 8.67T + 97T^{2} \)
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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

−15.43142502902488988984022138108, −14.43479911038056623326689917513, −13.18889941560067601811836412340, −12.13528905127250415362771128579, −10.77785264809635946130681871758, −9.747163547022751494433365317459, −8.986804939495419546454427947698, −6.23618779305855846614477458089, −4.63252960383607768840319553217, −3.18625564779949222248919522397, 3.49043408298866264078051416716, 6.08857269367458711966180965223, 6.60202840072830385175421894386, 8.192053342926042222931817011322, 9.376365413399842389583105946751, 11.55340427480765414470065605881, 12.73342264939743160497698077949, 13.63323563431591090795089569540, 14.38981626805694717126745140321, 15.90458583680815656398496261573

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