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Properties

Label 2-1064-7.2-c1-0-0
Degree $2$
Conductor $1064$
Sign $-0.676 - 0.736i$
Analytic cond. $8.49608$
Root an. cond. $2.91480$
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  + (−1.19 − 2.06i)3-s + (−1.62 + 2.81i)5-s + (2.05 − 1.66i)7-s + (−1.35 + 2.34i)9-s + (0.201 + 0.348i)11-s − 1.63·13-s + 7.77·15-s + (−2.21 − 3.84i)17-s + (−0.5 + 0.866i)19-s + (−5.90 − 2.27i)21-s + (−0.418 + 0.725i)23-s + (−2.78 − 4.82i)25-s − 0.686·27-s − 5.33·29-s + (−1.58 − 2.73i)31-s + ⋯
L(s)  = 1  + (−0.689 − 1.19i)3-s + (−0.727 + 1.25i)5-s + (0.777 − 0.628i)7-s + (−0.452 + 0.783i)9-s + (0.0607 + 0.105i)11-s − 0.452·13-s + 2.00·15-s + (−0.537 − 0.931i)17-s + (−0.114 + 0.198i)19-s + (−1.28 − 0.495i)21-s + (−0.0872 + 0.151i)23-s + (−0.557 − 0.965i)25-s − 0.132·27-s − 0.990·29-s + (−0.283 − 0.491i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $-0.676 - 0.736i$
Analytic conductor: \(8.49608\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1064} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1064,\ (\ :1/2),\ -0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07488883711\)
\(L(\frac12)\) \(\approx\) \(0.07488883711\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.05 + 1.66i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (1.19 + 2.06i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.62 - 2.81i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.201 - 0.348i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
17 \( 1 + (2.21 + 3.84i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.418 - 0.725i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.33T + 29T^{2} \)
31 \( 1 + (1.58 + 2.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.71 - 9.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 1.93T + 43T^{2} \)
47 \( 1 + (6.32 - 10.9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.75 - 9.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.58 + 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0632 - 0.109i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.23 - 2.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.17T + 71T^{2} \)
73 \( 1 + (6.25 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.20 + 3.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 + (7.29 - 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.50T + 97T^{2} \)
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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

−10.55781593724888766738658112740, −9.455233473291278663564001861273, −8.066217075149775658158071436443, −7.47700069140818611618555443169, −7.00261514240362325716977779906, −6.29619102562224780190840762634, −5.11698360282577094019323381486, −4.02960373531416957988607836827, −2.79169348136225241441780056816, −1.54197381377723814589403868840, 0.03652053292998746092628805865, 1.87923108333841110756599046221, 3.77189592705791715070181482157, 4.40156821228610395968867929771, 5.21566428271854983727431587800, 5.65248725476112437886966058579, 7.13793937403341477724673108306, 8.308606825210865479424113007211, 8.740013033303621031215133781745, 9.535561538382058809953215432649

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