Properties

Label 2-1064-7.4-c1-0-4
Degree $2$
Conductor $1064$
Sign $-0.971 - 0.236i$
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  + (−0.456 + 0.790i)3-s + (0.832 + 1.44i)5-s + (−0.462 + 2.60i)7-s + (1.08 + 1.87i)9-s + (0.121 − 0.209i)11-s − 5.76·13-s − 1.51·15-s + (−2.88 + 4.99i)17-s + (−0.5 − 0.866i)19-s + (−1.84 − 1.55i)21-s + (−3.30 − 5.72i)23-s + (1.11 − 1.92i)25-s − 4.71·27-s + 8.91·29-s + (2.96 − 5.13i)31-s + ⋯
L(s)  = 1  + (−0.263 + 0.456i)3-s + (0.372 + 0.644i)5-s + (−0.174 + 0.984i)7-s + (0.361 + 0.625i)9-s + (0.0365 − 0.0632i)11-s − 1.60·13-s − 0.392·15-s + (−0.699 + 1.21i)17-s + (−0.114 − 0.198i)19-s + (−0.403 − 0.339i)21-s + (−0.689 − 1.19i)23-s + (0.222 − 0.385i)25-s − 0.907·27-s + 1.65·29-s + (0.532 − 0.922i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $-0.971 - 0.236i$
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: \((2,\ 1064,\ (\ :1/2),\ -0.971 - 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9627097973\)
\(L(\frac12)\) \(\approx\) \(0.9627097973\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.462 - 2.60i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.456 - 0.790i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.832 - 1.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.121 + 0.209i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 + (2.88 - 4.99i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.30 + 5.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + (-2.96 + 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.90 - 8.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7.83T + 43T^{2} \)
47 \( 1 + (0.789 + 1.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.24 - 5.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.537 - 0.930i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.55 + 7.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.01 - 6.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.22T + 71T^{2} \)
73 \( 1 + (0.400 - 0.694i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.79 - 6.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.91T + 83T^{2} \)
89 \( 1 + (-6.55 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.11T + 97T^{2} \)
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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

−10.09368117962041174598469031253, −9.839910911125681532555025614578, −8.580891472248822716303919400220, −7.937826376831743267502671641616, −6.62211800073146860643915091861, −6.20792660904865438162488228945, −4.96381999299855453972478658401, −4.40735866508583561773998394602, −2.77075950276576421678093759202, −2.16002446331591563265131908488, 0.42622832103886750384284499703, 1.66484721875396313038981104333, 3.10676972109029697691392184153, 4.43049593174152534826150741863, 5.05214654620716800020098448905, 6.25862923006158128462522500156, 7.10856474711241594292223105165, 7.54671036579173586995457644710, 8.801413987103246605566253757848, 9.741340121703865168250851651084

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