Properties

Label 2-1064-7.4-c1-0-10
Degree $2$
Conductor $1064$
Sign $-0.0725 - 0.997i$
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  + (−0.292 + 0.507i)3-s + (0.5 + 0.866i)5-s + (2.62 − 0.358i)7-s + (1.32 + 2.30i)9-s + (−2.62 + 4.54i)11-s − 0.585·13-s − 0.585·15-s + (0.585 − 1.01i)17-s + (0.5 + 0.866i)19-s + (−0.585 + 1.43i)21-s + (−0.792 − 1.37i)23-s + (2 − 3.46i)25-s − 3.31·27-s − 6.58·29-s + (−3.70 + 6.42i)31-s + ⋯
L(s)  = 1  + (−0.169 + 0.292i)3-s + (0.223 + 0.387i)5-s + (0.990 − 0.135i)7-s + (0.442 + 0.766i)9-s + (−0.790 + 1.36i)11-s − 0.162·13-s − 0.151·15-s + (0.142 − 0.246i)17-s + (0.114 + 0.198i)19-s + (−0.127 + 0.313i)21-s + (−0.165 − 0.286i)23-s + (0.400 − 0.692i)25-s − 0.637·27-s − 1.22·29-s + (−0.665 + 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $-0.0725 - 0.997i$
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: \((2,\ 1064,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.575218138\)
\(L(\frac12)\) \(\approx\) \(1.575218138\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.358i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.292 - 0.507i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.62 - 4.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 + (-0.585 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.792 + 1.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 + (3.70 - 6.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.70 - 8.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 + (3.62 + 6.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.29 + 2.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.41 - 4.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.41 - 7.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.94 - 13.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.89T + 83T^{2} \)
89 \( 1 + (-8.77 - 15.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.65T + 97T^{2} \)
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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

−10.28905067682073897654539565709, −9.499503459946927173468711247838, −8.318350995788020175151921772307, −7.53930689244453877161762392190, −7.01803099030162402299017798211, −5.61773549903675269689541547992, −4.86986279534139366439163860987, −4.22531404265105460158925609637, −2.61045259508644669835326333224, −1.70008509013091594407758917150, 0.73820549249971382421961552798, 1.99254053155724850214341966848, 3.38621893714288368095051547640, 4.46797513248983616479128252634, 5.62893635821685918949489231156, 5.96365590029517939214210898831, 7.45668500928611116923566920325, 7.83890146650961482548349146127, 9.002273644206448533846929614431, 9.415146562757539915565636166603

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