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Properties

Label 2-1064-7.2-c1-0-24
Degree $2$
Conductor $1064$
Sign $-0.127 + 0.991i$
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  + (0.670 + 1.16i)3-s + (−1.45 + 2.51i)5-s + (−2.59 − 0.503i)7-s + (0.600 − 1.03i)9-s + (−2.49 − 4.32i)11-s + 1.55·13-s − 3.90·15-s + (−3.09 − 5.36i)17-s + (−0.5 + 0.866i)19-s + (−1.15 − 3.35i)21-s + (−3.39 + 5.87i)23-s + (−1.73 − 2.99i)25-s + 5.63·27-s + 0.707·29-s + (−4.78 − 8.28i)31-s + ⋯
L(s)  = 1  + (0.387 + 0.670i)3-s + (−0.650 + 1.12i)5-s + (−0.981 − 0.190i)7-s + (0.200 − 0.346i)9-s + (−0.752 − 1.30i)11-s + 0.430·13-s − 1.00·15-s + (−0.750 − 1.30i)17-s + (−0.114 + 0.198i)19-s + (−0.252 − 0.732i)21-s + (−0.707 + 1.22i)23-s + (−0.346 − 0.599i)25-s + 1.08·27-s + 0.131·29-s + (−0.859 − 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $-0.127 + 0.991i$
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: \((2,\ 1064,\ (\ :1/2),\ -0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4752332587\)
\(L(\frac12)\) \(\approx\) \(0.4752332587\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.503i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.670 - 1.16i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.45 - 2.51i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.49 + 4.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.55T + 13T^{2} \)
17 \( 1 + (3.09 + 5.36i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.39 - 5.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.707T + 29T^{2} \)
31 \( 1 + (4.78 + 8.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.12 - 3.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 1.25T + 43T^{2} \)
47 \( 1 + (-6.17 + 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.445 + 0.772i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.28 + 9.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.91 + 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.994 + 1.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (-3.50 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.46 - 14.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.494T + 83T^{2} \)
89 \( 1 + (-1.69 + 2.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−9.734594671346993508959355320333, −8.991904122597282752273076385246, −8.013175131997271344770866005494, −7.11939162554327845644199369661, −6.42110059358567760789077651586, −5.41852916305831710850731826485, −3.91039169795364291469489760026, −3.46405117750673936963513638159, −2.67908620975359977621217304080, −0.19632005506830970450591939028, 1.57325319904256154760127656288, 2.64178810241277432937885029862, 4.09933665446252785140637245070, 4.73141146684548291251792707453, 5.95668260425157006122689578799, 6.95991726941673679529071655304, 7.64419200627872036054613275161, 8.592684738335512965417882526183, 8.909801256649534797793543625534, 10.22595625594774138886140604693

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