Properties

Label 2-912-76.15-c1-0-16
Degree $2$
Conductor $912$
Sign $-0.672 + 0.740i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)3-s + (−1.93 + 1.62i)5-s + (−0.386 − 0.223i)7-s + (−0.939 + 0.342i)9-s + (−2.02 + 1.16i)11-s + (0.358 + 0.0632i)13-s + (−1.93 − 1.62i)15-s + (−5.58 − 2.03i)17-s + (−0.354 − 4.34i)19-s + (0.152 − 0.419i)21-s + (2.14 − 2.55i)23-s + (0.245 − 1.39i)25-s + (−0.5 − 0.866i)27-s + (0.449 + 1.23i)29-s + (3.35 − 5.80i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (−0.867 + 0.727i)5-s + (−0.146 − 0.0843i)7-s + (−0.313 + 0.114i)9-s + (−0.609 + 0.351i)11-s + (0.0994 + 0.0175i)13-s + (−0.500 − 0.420i)15-s + (−1.35 − 0.493i)17-s + (−0.0813 − 0.996i)19-s + (0.0333 − 0.0915i)21-s + (0.447 − 0.533i)23-s + (0.0490 − 0.278i)25-s + (−0.0962 − 0.166i)27-s + (0.0834 + 0.229i)29-s + (0.601 − 1.04i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.354 + 4.34i)T \)
good5 \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.386 + 0.223i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.358 - 0.0632i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (5.58 + 2.03i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.14 + 2.55i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.449 - 1.23i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.35 + 5.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.52iT - 37T^{2} \)
41 \( 1 + (11.2 - 1.99i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (4.52 + 5.39i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.66 + 7.33i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.13 - 1.34i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.69 - 2.07i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.19 - 4.35i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.71 - 3.17i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.11 - 3.45i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.591 + 3.35i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.442 + 2.50i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.39 + 0.802i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.14 + 1.08i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.368 + 1.01i)T + (-74.3 - 62.3i)T^{2} \)
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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

−9.969786130160871834538692497999, −8.871995998233588490799366179409, −8.214757328136465652881406242217, −7.08082586677646696753266020080, −6.64313153673585398895857140778, −5.12624894560113856133342796752, −4.40716727667572893556870344324, −3.33357606915733909208737220647, −2.44183304658287403772041005704, 0, 1.60153717205985677931377844336, 3.04934310921968617210941227109, 4.13410847826956390871703289852, 5.10987681480055099933915975474, 6.17410677997778884870963752208, 7.06475524252244403001340233127, 8.170969525023226694758043164972, 8.375307489315371126014184374377, 9.390209601293767978343409275530

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