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Properties

Label 2-1064-7.4-c1-0-13
Degree $2$
Conductor $1064$
Sign $0.928 - 0.371i$
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 + (1.13 + 2.38i)7-s + (−0.499 − 0.866i)9-s + (−1.13 + 1.97i)11-s + 2·13-s + 1.99·15-s + (−3.63 + 6.30i)17-s + (0.5 + 0.866i)19-s + (5.27 + 0.418i)21-s + (3.77 + 6.53i)23-s + (2 − 3.46i)25-s + 4.00·27-s − 4·29-s + (2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + (0.429 + 0.902i)7-s + (−0.166 − 0.288i)9-s + (−0.342 + 0.594i)11-s + 0.554·13-s + 0.516·15-s + (−0.882 + 1.52i)17-s + (0.114 + 0.198i)19-s + (1.15 + 0.0913i)21-s + (0.787 + 1.36i)23-s + (0.400 − 0.692i)25-s + 0.769·27-s − 0.742·29-s + (0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.928 - 0.371i$
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: \((2,\ 1064,\ (\ :1/2),\ 0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084941264\)
\(L(\frac12)\) \(\approx\) \(2.084941264\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 - 2.38i)T \)
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 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3.63 - 6.30i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.77 - 6.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-1.13 - 1.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.27 + 5.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.13 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (1.86 - 3.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.274 + 0.476i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (8.27 + 14.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.54T + 97T^{2} \)
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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

−9.878091536541258072277823929412, −8.861990321745235290632223200273, −8.272481267488446567658373671126, −7.53921933280499869396762476310, −6.64263002255482835675462554988, −5.88374779516592086345983494921, −4.80636586941462413634319296665, −3.45661572302742982376228686599, −2.24860615017460649403314278932, −1.67923524593089643365672993965, 0.928192953140979187610341207467, 2.69933173894425796717980523193, 3.64318409894071543183423344427, 4.63879686091347984091212209607, 5.14123563982487800681224342904, 6.58371535549691392393358800706, 7.37550858480586501013252029128, 8.594645967354352885021259570331, 8.919114582488040928971965732452, 9.787603877037124753149550073231

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