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Properties

Label 2-1071-17.4-c1-0-39
Degree $2$
Conductor $1071$
Sign $-0.853 - 0.520i$
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  − 1.20i·2-s + 0.539·4-s + (−2.18 + 2.18i)5-s + (−0.707 − 0.707i)7-s − 3.06i·8-s + (2.63 + 2.63i)10-s + (0.126 + 0.126i)11-s − 2.67·13-s + (−0.854 + 0.854i)14-s − 2.63·16-s + (−1.70 + 3.75i)17-s − 2.71i·19-s + (−1.17 + 1.17i)20-s + (0.153 − 0.153i)22-s + (−6.48 − 6.48i)23-s + ⋯
L(s)  = 1  − 0.854i·2-s + 0.269·4-s + (−0.975 + 0.975i)5-s + (−0.267 − 0.267i)7-s − 1.08i·8-s + (0.833 + 0.833i)10-s + (0.0382 + 0.0382i)11-s − 0.740·13-s + (−0.228 + 0.228i)14-s − 0.657·16-s + (−0.412 + 0.910i)17-s − 0.623i·19-s + (−0.262 + 0.262i)20-s + (0.0327 − 0.0327i)22-s + (−1.35 − 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (820, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2148725190\)
\(L(\frac12)\) \(\approx\) \(0.2148725190\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (1.70 - 3.75i)T \)
good2 \( 1 + 1.20iT - 2T^{2} \)
5 \( 1 + (2.18 - 2.18i)T - 5iT^{2} \)
11 \( 1 + (-0.126 - 0.126i)T + 11iT^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
19 \( 1 + 2.71iT - 19T^{2} \)
23 \( 1 + (6.48 + 6.48i)T + 23iT^{2} \)
29 \( 1 + (-4.28 + 4.28i)T - 29iT^{2} \)
31 \( 1 + (4.67 - 4.67i)T - 31iT^{2} \)
37 \( 1 + (7.70 - 7.70i)T - 37iT^{2} \)
41 \( 1 + (-1.27 - 1.27i)T + 41iT^{2} \)
43 \( 1 + 5.88iT - 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 5.01iT - 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 + (3.25 + 3.25i)T + 61iT^{2} \)
67 \( 1 + 3.27T + 67T^{2} \)
71 \( 1 + (3.18 - 3.18i)T - 71iT^{2} \)
73 \( 1 + (3.96 - 3.96i)T - 73iT^{2} \)
79 \( 1 + (1.07 + 1.07i)T + 79iT^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-6.01 + 6.01i)T - 97iT^{2} \)
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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

−9.853418130333695117510044245324, −8.543786795111673382001482999386, −7.69132409425834244136780129937, −6.77059596130395030779962017096, −6.41405537278229383963122925951, −4.66610746839847143704260297844, −3.75203045469072522503029354605, −3.00289543733799210175258436888, −1.99069559843530092995573608469, −0.086944412750277601850762329967, 1.90080099498484570827115627827, 3.33222941371334048635425476555, 4.51581054029604855108394088793, 5.32202060872748395945719405928, 6.13337611194649642965263292531, 7.32714088394698971390048689491, 7.65526280088334566265626815728, 8.572467023914739903783436837767, 9.249065120250965819745097203872, 10.26967377793509112790588138836

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