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Properties

Label 2-1248-104.101-c1-0-27
Degree $2$
Conductor $1248$
Sign $-0.476 + 0.879i$
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  + (0.866 − 0.5i)3-s + 1.88·5-s + (−3.80 − 2.19i)7-s + (0.499 − 0.866i)9-s + (0.0662 + 0.114i)11-s + (−3.59 + 0.274i)13-s + (1.63 − 0.942i)15-s + (1.14 − 1.98i)17-s + (2.18 − 3.79i)19-s − 4.39·21-s + (−1.97 − 3.42i)23-s − 1.44·25-s − 0.999i·27-s + (2.20 − 1.27i)29-s + 1.06i·31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + 0.843·5-s + (−1.43 − 0.831i)7-s + (0.166 − 0.288i)9-s + (0.0199 + 0.0346i)11-s + (−0.997 + 0.0760i)13-s + (0.421 − 0.243i)15-s + (0.278 − 0.482i)17-s + (0.502 − 0.870i)19-s − 0.959·21-s + (−0.411 − 0.713i)23-s − 0.288·25-s − 0.192i·27-s + (0.409 − 0.236i)29-s + 0.191i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405581810\)
\(L(\frac12)\) \(\approx\) \(1.405581810\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.59 - 0.274i)T \)
good5 \( 1 - 1.88T + 5T^{2} \)
7 \( 1 + (3.80 + 2.19i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0662 - 0.114i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.14 + 1.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.18 + 3.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.97 + 3.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.20 + 1.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.06iT - 31T^{2} \)
37 \( 1 + (5.28 + 9.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.14 + 4.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.32 + 4.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + 8.28iT - 53T^{2} \)
59 \( 1 + (1.65 - 2.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.40 - 3.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.540 + 0.936i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.45 + 3.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.14iT - 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + 0.144T + 83T^{2} \)
89 \( 1 + (14.2 - 8.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.11 - 3.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.598363078295095581180229534383, −8.839865730714462066125855258824, −7.55948732408693384130427972946, −7.00582967912905934989688239161, −6.28682670815400400286891259226, −5.26582523220374170122020726509, −4.07794916424204595856152411673, −3.04671758124748820435962646183, −2.20265783259929297518024309619, −0.51762848052672618243513333388, 1.83630959373039190520584658374, 2.87639179263667039938538269002, 3.59811438518765936031948687669, 5.01300855433414443380328423212, 5.85341299965750960549310814899, 6.48490769606908815037346683176, 7.57400521747662539585538972227, 8.505192239104365498745071569640, 9.378756705011414415085269986195, 9.961687404130500605579015408221

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