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Properties

Label 2-1071-17.16-c1-0-18
Degree $2$
Conductor $1071$
Sign $0.787 - 0.615i$
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  + 1.17·2-s − 0.630·4-s − 0.460i·5-s + i·7-s − 3.07·8-s − 0.539i·10-s + 0.709i·11-s + 3.87·13-s + 1.17i·14-s − 2.34·16-s + (2.53 + 3.24i)17-s + 4.17·19-s + 0.290i·20-s + 0.829i·22-s + 7.34i·23-s + ⋯
L(s)  = 1  + 0.827·2-s − 0.315·4-s − 0.206i·5-s + 0.377i·7-s − 1.08·8-s − 0.170i·10-s + 0.213i·11-s + 1.07·13-s + 0.312i·14-s − 0.585·16-s + (0.615 + 0.787i)17-s + 0.956·19-s + 0.0650i·20-s + 0.176i·22-s + 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $0.787 - 0.615i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ 0.787 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.093398382\)
\(L(\frac12)\) \(\approx\) \(2.093398382\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
17 \( 1 + (-2.53 - 3.24i)T \)
good2 \( 1 - 1.17T + 2T^{2} \)
5 \( 1 + 0.460iT - 5T^{2} \)
11 \( 1 - 0.709iT - 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 - 7.34iT - 23T^{2} \)
29 \( 1 - 1.70iT - 29T^{2} \)
31 \( 1 + 4.87iT - 31T^{2} \)
37 \( 1 - 1.90iT - 37T^{2} \)
41 \( 1 - 4.17iT - 41T^{2} \)
43 \( 1 - 8.75T + 43T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 + 4.58T + 53T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 6.81T + 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 - 5.23iT - 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 - 0.496T + 83T^{2} \)
89 \( 1 + 3.60T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−9.814564866335328236072856262610, −9.179252617283917175686019758977, −8.391006826100765450017926351169, −7.49981935167957437677644736682, −6.20375429170005546671139511471, −5.65117581188357497228853787096, −4.79664128491533734052276120189, −3.76732449024101592478269504015, −3.04195157276556977846671886636, −1.32797406504541763915716296272, 0.861050433460864711272302440610, 2.82458307650704308460410803245, 3.58614034699358291296482786655, 4.56399506519291818741416860725, 5.38581371108785572218686639199, 6.26798497517249938018296995432, 7.10517369893132013629694534737, 8.235085738606234473649770263428, 8.945814594287431762387389455932, 9.801605576018480040777898336577

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