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Properties

Label 2-48-16.5-c1-0-1
Degree $2$
Conductor $48$
Sign $0.995 - 0.0985i$
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  + (−1.34 + 0.443i)2-s + (0.707 − 0.707i)3-s + (1.60 − 1.19i)4-s + (1.27 + 1.27i)5-s + (−0.635 + 1.26i)6-s − 0.158i·7-s + (−1.62 + 2.31i)8-s − 1.00i·9-s + (−2.27 − 1.14i)10-s + (−3.79 − 3.79i)11-s + (0.292 − 1.97i)12-s + (−4.21 + 4.21i)13-s + (0.0705 + 0.213i)14-s + 1.79·15-s + (1.15 − 3.82i)16-s + 3.05·17-s + ⋯
L(s)  = 1  + (−0.949 + 0.313i)2-s + (0.408 − 0.408i)3-s + (0.803 − 0.595i)4-s + (0.568 + 0.568i)5-s + (−0.259 + 0.515i)6-s − 0.0600i·7-s + (−0.575 + 0.817i)8-s − 0.333i·9-s + (−0.718 − 0.361i)10-s + (−1.14 − 1.14i)11-s + (0.0845 − 0.571i)12-s + (−1.16 + 1.16i)13-s + (0.0188 + 0.0570i)14-s + 0.464·15-s + (0.289 − 0.957i)16-s + 0.740·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.995 - 0.0985i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.995 - 0.0985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.645225 + 0.0318835i\)
\(L(\frac12)\) \(\approx\) \(0.645225 + 0.0318835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.443i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-1.27 - 1.27i)T + 5iT^{2} \)
7 \( 1 + 0.158iT - 7T^{2} \)
11 \( 1 + (3.79 + 3.79i)T + 11iT^{2} \)
13 \( 1 + (4.21 - 4.21i)T - 13iT^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + (2.15 - 2.15i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-2.09 + 2.09i)T - 29iT^{2} \)
31 \( 1 - 4.15T + 31T^{2} \)
37 \( 1 + (5.98 + 5.98i)T + 37iT^{2} \)
41 \( 1 + 2.60iT - 41T^{2} \)
43 \( 1 + (-5.75 - 5.75i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-3.55 - 3.55i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (-3.66 + 3.66i)T - 61iT^{2} \)
67 \( 1 + (-0.767 + 0.767i)T - 67iT^{2} \)
71 \( 1 - 0.317iT - 71T^{2} \)
73 \( 1 + 1.33iT - 73T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + (-0.115 + 0.115i)T - 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−15.88964805636840472698189183880, −14.53297203486061588609015430457, −13.82099296280306575777526224297, −12.08375113252286482808334836731, −10.67469645814641764617262851608, −9.671708471960232416786633649342, −8.305522493472280612478009275388, −7.14995200691863597812186463687, −5.81495267194521250421158340232, −2.49704459496081741143324106588, 2.58012679175009916972193838344, 5.11635332520457461218234470740, 7.36084009320182875941460385472, 8.517736335457635755693965821627, 9.883707634985119969315230270295, 10.36853297439090355671021383802, 12.27148432817518999734011634912, 13.07398197365761648406200420026, 14.91568359895856198804697804233, 15.74639593513314400293128352482

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