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Properties

Label 2-1064-7.2-c1-0-15
Degree $2$
Conductor $1064$
Sign $0.386 + 0.922i$
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 + (−2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 4·13-s + 1.99·15-s + (−1 − 1.73i)17-s + (−0.5 + 0.866i)19-s + (4 + 3.46i)21-s + (3.5 − 6.06i)23-s + (2 + 3.46i)25-s − 4.00·27-s + 2·29-s + (−3 − 5.19i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (−0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 1.10·13-s + 0.516·15-s + (−0.242 − 0.420i)17-s + (−0.114 + 0.198i)19-s + (0.872 + 0.755i)21-s + (0.729 − 1.26i)23-s + (0.400 + 0.692i)25-s − 0.769·27-s + 0.371·29-s + (−0.538 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.386 + 0.922i$
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: \((2,\ 1064,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.116174256\)
\(L(\frac12)\) \(\approx\) \(1.116174256\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
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 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−9.542237632783190363110083671897, −9.096827297599335463506542525907, −7.81591177673674477281837322463, −7.07209841530330156611002946338, −6.38377505530515802809230800678, −5.89564055134597059218867911319, −4.45532640588309260919232421281, −3.36151952398278646930131537947, −2.14700054113443752242126418183, −0.70221079048893333270470609790, 1.04145490339913171875300757118, 3.14971981481030273137137794166, 3.91975547138372311145867777825, 4.73060125098432165301508575413, 5.84008536742768941390217106627, 6.39389551720631166389588537865, 7.55755532942875722679776167276, 8.728276234555995029550995546085, 9.203921037253023897027227078430, 10.16520660527086228687818176258

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