Properties

Label 2-48-48.35-c1-0-5
Degree $2$
Conductor $48$
Sign $0.176 + 0.984i$
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  + (0.416 − 1.35i)2-s + (−1.43 − 0.966i)3-s + (−1.65 − 1.12i)4-s + (1.57 + 1.57i)5-s + (−1.90 + 1.54i)6-s + 2.24·7-s + (−2.20 + 1.76i)8-s + (1.13 + 2.77i)9-s + (2.77 − 1.47i)10-s + (−1.13 + 1.13i)11-s + (1.29 + 3.21i)12-s + (−3.24 − 3.24i)13-s + (0.935 − 3.04i)14-s + (−0.739 − 3.77i)15-s + (1.47 + 3.71i)16-s + 1.66i·17-s + ⋯
L(s)  = 1  + (0.294 − 0.955i)2-s + (−0.829 − 0.558i)3-s + (−0.826 − 0.562i)4-s + (0.702 + 0.702i)5-s + (−0.777 + 0.628i)6-s + 0.850·7-s + (−0.780 + 0.624i)8-s + (0.377 + 0.926i)9-s + (0.878 − 0.465i)10-s + (−0.341 + 0.341i)11-s + (0.372 + 0.928i)12-s + (−0.901 − 0.901i)13-s + (0.250 − 0.812i)14-s + (−0.191 − 0.975i)15-s + (0.367 + 0.929i)16-s + 0.403i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599699 - 0.501865i\)
\(L(\frac12)\) \(\approx\) \(0.599699 - 0.501865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.416 + 1.35i)T \)
3 \( 1 + (1.43 + 0.966i)T \)
good5 \( 1 + (-1.57 - 1.57i)T + 5iT^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + (1.13 - 1.13i)T - 11iT^{2} \)
13 \( 1 + (3.24 + 3.24i)T + 13iT^{2} \)
17 \( 1 - 1.66iT - 17T^{2} \)
19 \( 1 + (3.77 - 3.77i)T - 19iT^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + (-3.23 + 3.23i)T - 29iT^{2} \)
31 \( 1 - 1.30iT - 31T^{2} \)
37 \( 1 + (-2.30 + 2.30i)T - 37iT^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + (-3.77 - 3.77i)T + 43iT^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 + (0.972 + 0.972i)T + 53iT^{2} \)
59 \( 1 + (3.88 - 3.88i)T - 59iT^{2} \)
61 \( 1 + (-4.19 - 4.19i)T + 61iT^{2} \)
67 \( 1 + (-8.02 + 8.02i)T - 67iT^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + 6.38iT - 73T^{2} \)
79 \( 1 - 2.69iT - 79T^{2} \)
83 \( 1 + (2.61 + 2.61i)T + 83iT^{2} \)
89 \( 1 - 7.35T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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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

−15.03107225786769395318045495470, −14.10739171418719838041759868407, −12.89197755385284485518647476138, −12.01781200717581045087508472205, −10.68776409511226974382761045146, −10.16134878515106964803907815814, −8.008315207075058814838626075355, −6.16292571127716224411723999073, −4.86923829016358286536353615561, −2.19759953500604809690517725123, 4.60154615981983080022896872675, 5.39287300666777092378821769418, 6.89700492669798142989768056944, 8.681104814178847815338033007124, 9.752725871091366094722358297328, 11.42234074808702762056818484713, 12.64102932762574931537437456828, 13.85157533333164765927770932037, 14.97890021226482603588154770718, 16.06076839313560804303343518104

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