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Properties

Label 2-48-48.11-c1-0-1
Degree $2$
Conductor $48$
Sign $0.601 - 0.798i$
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 + (1.73 + 0.0835i)3-s + (−1.26 − 1.55i)4-s + (−0.431 + 0.431i)5-s + (−1.15 + 2.15i)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 + (−2.05 − 2.78i)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 + (0.998 + 0.0482i)3-s + (−0.631 − 0.775i)4-s + (−0.193 + 0.193i)5-s + (−0.472 + 0.881i)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 + (−0.592 − 0.805i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.601 - 0.798i$
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: \((2,\ 48,\ (\ :1/2),\ 0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689590 + 0.343805i\)
\(L(\frac12)\) \(\approx\) \(0.689590 + 0.343805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0835i)T \)
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

−15.65841891711355248523072162619, −15.14150522741052384097541751391, −13.55779622902372501817095490499, −13.13763602940720641469581619741, −10.68129960931985490636996949661, −9.579012256969933180369606190462, −8.443458162243271684656364578110, −7.35382845817678160785451825516, −5.83189739480095664314390603882, −3.49868648684200611022182584110, 2.62921195346481332736716364162, 4.22408828721365359568982694189, 7.12988751506212690700357212695, 8.511286881223734788438966402518, 9.552919828811655588687381678828, 10.49861750101615485450964118617, 12.33876489375447525252686243820, 13.00094794679960979468108892038, 14.09044800316543434846491203469, 15.67038653986214091428906714891

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