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Properties

Label 2-1064-7.4-c1-0-20
Degree $2$
Conductor $1064$
Sign $0.749 + 0.661i$
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  + (−1.70 + 2.95i)3-s + (0.5 + 0.866i)5-s + (−1.62 + 2.09i)7-s + (−4.32 − 7.49i)9-s + (1.62 − 2.80i)11-s − 3.41·13-s − 3.41·15-s + (3.41 − 5.91i)17-s + (0.5 + 0.866i)19-s + (−3.41 − 8.36i)21-s + (−2.20 − 3.82i)23-s + (2 − 3.46i)25-s + 19.3·27-s − 9.41·29-s + (−2.29 + 3.97i)31-s + ⋯
L(s)  = 1  + (−0.985 + 1.70i)3-s + (0.223 + 0.387i)5-s + (−0.612 + 0.790i)7-s + (−1.44 − 2.49i)9-s + (0.488 − 0.846i)11-s − 0.946·13-s − 0.881·15-s + (0.828 − 1.43i)17-s + (0.114 + 0.198i)19-s + (−0.745 − 1.82i)21-s + (−0.460 − 0.797i)23-s + (0.400 − 0.692i)25-s + 3.71·27-s − 1.74·29-s + (−0.411 + 0.713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.749 + 0.661i$
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: \((2,\ 1064,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3775548823\)
\(L(\frac12)\) \(\approx\) \(0.3775548823\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.09i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.70 - 2.95i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.62 + 2.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.20 + 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.41T + 29T^{2} \)
31 \( 1 + (2.29 - 3.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.29 - 5.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + (-0.621 - 1.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.70 + 4.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.414 + 0.717i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.58 - 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.94 + 3.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (6.77 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.867047418691042556214531019196, −9.318894748695925655623344491155, −8.550431195621014982667948254297, −6.96716376496831847103488813636, −6.10882813350768036544735164639, −5.46169417899628318250960145893, −4.73575006894212691377700871312, −3.52937783438264992911283403909, −2.85026177629041115015121060093, −0.20312103242461230923494069375, 1.26558599400264608110830994744, 2.10608580937985602399105759901, 3.80075305132301457261347067181, 5.16919536432140261210541729528, 5.84868947298499958039856035081, 6.71720122548997463406120638327, 7.45009673958839532571508670484, 7.81939845644147666451160274222, 9.216524522800134709872859169306, 10.08429716145478908510696577707

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