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Properties

Label 2-1071-7.4-c1-0-22
Degree $2$
Conductor $1071$
Sign $0.386 - 0.922i$
Analytic cond. $8.55197$
Root an. cond. $2.92437$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−2 + 1.73i)7-s + 3·8-s + (−1.5 + 2.59i)10-s + (3 − 5.19i)11-s + 13-s + (−2.5 − 0.866i)14-s + (0.500 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (2 + 3.46i)19-s + 3·20-s + 6·22-s + (2 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s + 1.06·8-s + (−0.474 + 0.821i)10-s + (0.904 − 1.56i)11-s + 0.277·13-s + (−0.668 − 0.231i)14-s + (0.125 + 0.216i)16-s + (−0.121 + 0.210i)17-s + (0.458 + 0.794i)19-s + 0.670·20-s + 1.27·22-s + (0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479321056\)
\(L(\frac12)\) \(\approx\) \(2.479321056\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.12237919082952657319622433503, −9.257128849302038995483283601708, −8.393056621694735950159986182568, −7.11728869835905466669126672404, −6.47281252993319693230881983875, −5.98594335069302067964830398988, −5.33767484341052500650891695341, −3.65890322653304908679200282535, −2.90577556172763788980181151164, −1.49919855892003166702434593954, 1.15807201173852546747242214322, 2.22103824551955356977735315487, 3.48383452675271022971091152555, 4.47933589507849181220015900441, 5.00556484727790607526042897580, 6.59957024545222033655778502661, 6.99493873357664471222860795373, 8.165829775413167881415288830911, 9.141596268552397601724739856327, 9.767731668144289755730142869259

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