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Properties

Label 2-1248-104.99-c1-0-18
Degree $2$
Conductor $1248$
Sign $0.990 - 0.139i$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  + 3-s + (1.98 − 1.98i)5-s + (3.05 + 3.05i)7-s + 9-s + (1.08 − 1.08i)11-s + (−3.57 + 0.429i)13-s + (1.98 − 1.98i)15-s + 5.70i·17-s + (4.39 + 4.39i)19-s + (3.05 + 3.05i)21-s + 2.95·23-s − 2.84i·25-s + 27-s − 6.96i·29-s + (−3.05 + 3.05i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.885 − 0.885i)5-s + (1.15 + 1.15i)7-s + 0.333·9-s + (0.327 − 0.327i)11-s + (−0.992 + 0.119i)13-s + (0.511 − 0.511i)15-s + 1.38i·17-s + (1.00 + 1.00i)19-s + (0.667 + 0.667i)21-s + 0.615·23-s − 0.569i·25-s + 0.192·27-s − 1.29i·29-s + (−0.549 + 0.549i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.695556939\)
\(L(\frac12)\) \(\approx\) \(2.695556939\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 + (3.57 - 0.429i)T \)
good5 \( 1 + (-1.98 + 1.98i)T - 5iT^{2} \)
7 \( 1 + (-3.05 - 3.05i)T + 7iT^{2} \)
11 \( 1 + (-1.08 + 1.08i)T - 11iT^{2} \)
17 \( 1 - 5.70iT - 17T^{2} \)
19 \( 1 + (-4.39 - 4.39i)T + 19iT^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 + 6.96iT - 29T^{2} \)
31 \( 1 + (3.05 - 3.05i)T - 31iT^{2} \)
37 \( 1 + (5.14 + 5.14i)T + 37iT^{2} \)
41 \( 1 + (2.77 + 2.77i)T + 41iT^{2} \)
43 \( 1 - 3.00iT - 43T^{2} \)
47 \( 1 + (6.40 + 6.40i)T + 47iT^{2} \)
53 \( 1 + 4.28iT - 53T^{2} \)
59 \( 1 + (3.00 - 3.00i)T - 59iT^{2} \)
61 \( 1 - 1.13iT - 61T^{2} \)
67 \( 1 + (7.39 + 7.39i)T + 67iT^{2} \)
71 \( 1 + (-3.20 + 3.20i)T - 71iT^{2} \)
73 \( 1 + (-7.87 + 7.87i)T - 73iT^{2} \)
79 \( 1 + 4.46iT - 79T^{2} \)
83 \( 1 + (-9.78 - 9.78i)T + 83iT^{2} \)
89 \( 1 + (-4.24 + 4.24i)T - 89iT^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−9.507868568823121994473397049657, −8.875492138365261622977041149724, −8.298231110726326819622321696936, −7.53631652177494933318023338631, −6.15819731892827321680762614910, −5.39965394707777385118866237005, −4.81836727987360986712797462092, −3.52574633322146833311237546260, −2.09693664554322259915750725085, −1.58162946526905246480018974774, 1.26979577037839984271566684327, 2.45573451549901257486702449542, 3.30866010936863580495402822429, 4.71606106332859354258641869232, 5.16491228383053170876021382826, 6.81163741405332141881058130210, 7.14896594158160238163586379947, 7.82439730457682644171906613874, 9.071721146179939481771643336062, 9.692564659652811733040895579782

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