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Properties

Label 2-1071-7.4-c1-0-8
Degree $2$
Conductor $1071$
Sign $-0.200 + 0.979i$
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  + (−1.24 − 2.15i)2-s + (−2.10 + 3.63i)4-s + (−1.16 − 2.01i)5-s + (−1.62 + 2.08i)7-s + 5.47·8-s + (−2.90 + 5.02i)10-s + (−2.57 + 4.45i)11-s − 0.596·13-s + (6.52 + 0.908i)14-s + (−2.62 − 4.53i)16-s + (0.5 − 0.866i)17-s + (−0.946 − 1.63i)19-s + 9.78·20-s + 12.8·22-s + (−2.07 − 3.60i)23-s + ⋯
L(s)  = 1  + (−0.880 − 1.52i)2-s + (−1.05 + 1.81i)4-s + (−0.521 − 0.902i)5-s + (−0.614 + 0.788i)7-s + 1.93·8-s + (−0.917 + 1.58i)10-s + (−0.776 + 1.34i)11-s − 0.165·13-s + (1.74 + 0.242i)14-s + (−0.655 − 1.13i)16-s + (0.121 − 0.210i)17-s + (−0.217 − 0.376i)19-s + 2.18·20-s + 2.73·22-s + (−0.433 − 0.751i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $-0.200 + 0.979i$
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: \((2,\ 1071,\ (\ :1/2),\ -0.200 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5934803839\)
\(L(\frac12)\) \(\approx\) \(0.5934803839\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.08i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.24 + 2.15i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.57 - 4.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.596T + 13T^{2} \)
19 \( 1 + (0.946 + 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.07 + 3.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 + (-3.22 + 5.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.70 - 8.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.09T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-2.19 - 3.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.46 - 9.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.60 + 2.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.29 + 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.38 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.91T + 71T^{2} \)
73 \( 1 + (0.217 - 0.375i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.07 - 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.17T + 83T^{2} \)
89 \( 1 + (-0.197 - 0.341i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−9.598283947691153166796910206021, −9.202321883807680106661680642002, −8.203705537602124224162908675629, −7.74786384805807288479872424016, −6.32498904552276213925804492980, −4.85001982761118874511474497731, −4.25229061855580700218373571130, −2.84895499111463121492772848267, −2.19042767696290618600688105070, −0.63825950737059363508249497136, 0.67435601592590439399319726159, 2.98654544285264963923317638199, 4.02961017955628957583084351412, 5.46854400016804367800354272126, 6.14714129952695311292495709066, 6.95842322781815647001735088711, 7.59556946725604544295675800365, 8.186664894195689605164869248220, 9.080171181034915091005204946953, 10.03857080471600358259203511593

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