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Properties

Label 2-1064-7.4-c1-0-28
Degree $2$
Conductor $1064$
Sign $0.0158 + 0.999i$
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 − 1.73i)3-s + (0.5 + 0.866i)5-s + (−2.63 + 0.209i)7-s + (−0.499 − 0.866i)9-s + (2.63 − 4.56i)11-s + 2·13-s + 1.99·15-s + (0.137 − 0.238i)17-s + (0.5 + 0.866i)19-s + (−2.27 + 4.77i)21-s + (−3.77 − 6.53i)23-s + (2 − 3.46i)25-s + 4.00·27-s − 4·29-s + (2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + (−0.996 + 0.0791i)7-s + (−0.166 − 0.288i)9-s + (0.795 − 1.37i)11-s + 0.554·13-s + 0.516·15-s + (0.0333 − 0.0577i)17-s + (0.114 + 0.198i)19-s + (−0.496 + 1.04i)21-s + (−0.787 − 1.36i)23-s + (0.400 − 0.692i)25-s + 0.769·27-s − 0.742·29-s + (0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0158 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.0158 + 0.999i$
Analytic conductor: \(8.49608\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1064} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1064,\ (\ :1/2),\ 0.0158 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.855770646\)
\(L(\frac12)\) \(\approx\) \(1.855770646\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.209i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-0.137 + 0.238i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.77 + 6.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (2.63 + 4.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.27 - 7.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 + (5.63 - 9.76i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.27 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.54T + 83T^{2} \)
89 \( 1 + (0.725 + 1.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.54T + 97T^{2} \)
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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

−9.568217847547884872205783505372, −8.698144765581706308507872752809, −8.138222728463547493486893033694, −7.08479581917896327020878060820, −6.31895153814960150329396287190, −5.91120939301625435549577152616, −4.13157192260312719368471086254, −3.15102826234998405951016061405, −2.29534229269438225968543057917, −0.824328819533419104251203653337, 1.57393192133919303911306584758, 3.11121969107086715772331047243, 3.88770369929902212511977441208, 4.64925042422061066556007364240, 5.79351836519151312205487361284, 6.75780030578943437045825921320, 7.61893626560284621043583730102, 8.877446757272978260811668057538, 9.414767909261685303671591597977, 9.770438255361403423785469700050

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