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Properties

Label 2-48-48.35-c1-0-0
Degree $2$
Conductor $48$
Sign $0.627 - 0.778i$
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  + (−1.39 − 0.204i)2-s + (−0.814 + 1.52i)3-s + (1.91 + 0.573i)4-s + (2.08 + 2.08i)5-s + (1.45 − 1.97i)6-s − 1.14·7-s + (−2.56 − 1.19i)8-s + (−1.67 − 2.48i)9-s + (−2.48 − 3.34i)10-s + (1.67 − 1.67i)11-s + (−2.43 + 2.46i)12-s + (0.146 + 0.146i)13-s + (1.60 + 0.234i)14-s + (−4.88 + 1.48i)15-s + (3.34 + 2.19i)16-s − 5.59i·17-s + ⋯
L(s)  = 1  + (−0.989 − 0.144i)2-s + (−0.470 + 0.882i)3-s + (0.958 + 0.286i)4-s + (0.931 + 0.931i)5-s + (0.592 − 0.805i)6-s − 0.433·7-s + (−0.906 − 0.422i)8-s + (−0.558 − 0.829i)9-s + (−0.787 − 1.05i)10-s + (0.504 − 0.504i)11-s + (−0.703 + 0.710i)12-s + (0.0405 + 0.0405i)13-s + (0.428 + 0.0627i)14-s + (−1.26 + 0.384i)15-s + (0.835 + 0.549i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.627 - 0.778i$
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: \((2,\ 48,\ (\ :1/2),\ 0.627 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489378 + 0.234233i\)
\(L(\frac12)\) \(\approx\) \(0.489378 + 0.234233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.204i)T \)
3 \( 1 + (0.814 - 1.52i)T \)
good5 \( 1 + (-2.08 - 2.08i)T + 5iT^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 + (-1.67 + 1.67i)T - 11iT^{2} \)
13 \( 1 + (-0.146 - 0.146i)T + 13iT^{2} \)
17 \( 1 + 5.59iT - 17T^{2} \)
19 \( 1 + (-1.48 + 1.48i)T - 19iT^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + (3.51 - 3.51i)T - 29iT^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 - 0.610T + 41T^{2} \)
43 \( 1 + (1.48 + 1.48i)T + 43iT^{2} \)
47 \( 1 - 6.41T + 47T^{2} \)
53 \( 1 + (-0.164 - 0.164i)T + 53iT^{2} \)
59 \( 1 + (9.05 - 9.05i)T - 59iT^{2} \)
61 \( 1 + (-4.53 - 4.53i)T + 61iT^{2} \)
67 \( 1 + (0.635 - 0.635i)T - 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 9.83iT - 79T^{2} \)
83 \( 1 + (8.09 + 8.09i)T + 83iT^{2} \)
89 \( 1 - 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−16.11857758571428158175444325836, −15.04570204789720383423722336662, −13.76223171002311163171837455596, −11.80782650961055644880870632247, −10.89996512231204696049612845669, −9.839330008068927701384528138533, −9.141657938539346332277417531504, −6.99389724778086933820593516998, −5.83526797377675345309218897203, −3.10481947319434072200526019304, 1.69578359853725675531896880474, 5.63088384160470586525400218939, 6.69955058569404092588748504954, 8.235920566746012977433715157663, 9.404737451782279877588433927233, 10.63361851716249816660754198814, 12.17940975584803982055138831264, 12.93268888283903553826811291185, 14.38784783441989261318263004208, 16.04152980899198313282517734703

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