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Properties

Label 2-1064-7.2-c1-0-22
Degree $2$
Conductor $1064$
Sign $0.849 - 0.527i$
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  + (1.23 + 2.13i)3-s + (2.01 − 3.49i)5-s + (1.24 + 2.33i)7-s + (−1.55 + 2.68i)9-s + (−0.801 − 1.38i)11-s + 3.15·13-s + 9.97·15-s + (−0.326 − 0.565i)17-s + (−0.5 + 0.866i)19-s + (−3.44 + 5.55i)21-s + (−1.45 + 2.52i)23-s + (−5.65 − 9.79i)25-s − 0.248·27-s + 2.12·29-s + (3.51 + 6.08i)31-s + ⋯
L(s)  = 1  + (0.713 + 1.23i)3-s + (0.903 − 1.56i)5-s + (0.472 + 0.881i)7-s + (−0.516 + 0.895i)9-s + (−0.241 − 0.418i)11-s + 0.874·13-s + 2.57·15-s + (−0.0791 − 0.137i)17-s + (−0.114 + 0.198i)19-s + (−0.751 + 1.21i)21-s + (−0.304 + 0.526i)23-s + (−1.13 − 1.95i)25-s − 0.0477·27-s + 0.393·29-s + (0.630 + 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.849 - 0.527i$
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: \((2,\ 1064,\ (\ :1/2),\ 0.849 - 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.581840869\)
\(L(\frac12)\) \(\approx\) \(2.581840869\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 2.33i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.23 - 2.13i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.01 + 3.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.801 + 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 + (0.326 + 0.565i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.45 - 2.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + (-3.51 - 6.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.65 + 4.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 7.97T + 43T^{2} \)
47 \( 1 + (-0.302 + 0.524i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.24 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.22 + 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.67 - 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.39 - 2.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 + (-8.31 - 14.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.42 - 4.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 + (-1.13 + 1.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.24T + 97T^{2} \)
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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

−9.687446819174227736458607211354, −9.123778054072529355371298385274, −8.562859165618037542536597684737, −8.095863116130760731509830603180, −6.21575877720483154605905586578, −5.41105243406488213924561037706, −4.82255782769722886906743467420, −3.89718497538985769305831708130, −2.65476326352645307017976189434, −1.41575763431099396657306596658, 1.38311848132481886243977367099, 2.34753363972022806701986648447, 3.12917067083486825359143670759, 4.42868401314924873856329807611, 6.12560446818227445908505864523, 6.46380781956500604390922981416, 7.46142024238169136594755727016, 7.77765777311530955858909776968, 8.890753312338332146254247043284, 9.962938544944255774500301305986

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