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Properties

Label 2-1064-7.4-c1-0-25
Degree $2$
Conductor $1064$
Sign $0.727 + 0.686i$
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  + (0.218 − 0.378i)3-s + (0.149 + 0.259i)5-s + (1.68 − 2.03i)7-s + (1.40 + 2.43i)9-s + (2.59 − 4.49i)11-s − 1.26·13-s + 0.130·15-s + (−0.979 + 1.69i)17-s + (−0.5 − 0.866i)19-s + (−0.400 − 1.08i)21-s + (0.745 + 1.29i)23-s + (2.45 − 4.25i)25-s + 2.53·27-s − 0.530·29-s + (−0.974 + 1.68i)31-s + ⋯
L(s)  = 1  + (0.126 − 0.218i)3-s + (0.0670 + 0.116i)5-s + (0.638 − 0.769i)7-s + (0.468 + 0.811i)9-s + (0.782 − 1.35i)11-s − 0.349·13-s + 0.0337·15-s + (−0.237 + 0.411i)17-s + (−0.114 − 0.198i)19-s + (−0.0874 − 0.236i)21-s + (0.155 + 0.269i)23-s + (0.491 − 0.850i)25-s + 0.488·27-s − 0.0984·29-s + (−0.174 + 0.303i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.727 + 0.686i$
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: \((2,\ 1064,\ (\ :1/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927281035\)
\(L(\frac12)\) \(\approx\) \(1.927281035\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 2.03i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.218 + 0.378i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.149 - 0.259i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 + (0.979 - 1.69i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.745 - 1.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.530T + 29T^{2} \)
31 \( 1 + (0.974 - 1.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.09 + 3.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + (-4.27 - 7.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.87 + 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.40 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.96 - 5.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.75 - 4.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.28T + 71T^{2} \)
73 \( 1 + (-2.34 + 4.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.84 + 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + (-1.92 - 3.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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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

−9.897345468832243085902842340845, −8.773221475440651199973055746072, −8.187131235235316941714255250411, −7.29525164589083228967777713842, −6.57494887056163268702128322074, −5.47731926282073445678293730257, −4.48782672592937526939683406443, −3.60551622545173590684452558860, −2.23782482269404314430288415612, −0.992980007023406893000022249582, 1.41928720539049182698173968072, 2.55177856804825837043226157264, 3.94886766923264600445971930567, 4.70041835455994763064702763794, 5.62816927847138125252300956254, 6.79892292563886496182003231346, 7.34599697872200929337969403708, 8.574261574866351276180965348719, 9.232348857782233338395101712206, 9.772789223035116280111079075968

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