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Properties

Label 2-1071-17.4-c1-0-19
Degree $2$
Conductor $1071$
Sign $0.481 - 0.876i$
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  + 0.569i·2-s + 1.67·4-s + (−0.119 + 0.119i)5-s + (−0.707 − 0.707i)7-s + 2.09i·8-s + (−0.0681 − 0.0681i)10-s + (2.80 + 2.80i)11-s + 1.71·13-s + (0.403 − 0.403i)14-s + 2.15·16-s + (−3.82 − 1.52i)17-s + 2.80i·19-s + (−0.200 + 0.200i)20-s + (−1.59 + 1.59i)22-s + (1.19 + 1.19i)23-s + ⋯
L(s)  = 1  + 0.403i·2-s + 0.837·4-s + (−0.0534 + 0.0534i)5-s + (−0.267 − 0.267i)7-s + 0.740i·8-s + (−0.0215 − 0.0215i)10-s + (0.845 + 0.845i)11-s + 0.476·13-s + (0.107 − 0.107i)14-s + 0.539·16-s + (−0.928 − 0.371i)17-s + 0.643i·19-s + (−0.0447 + 0.0447i)20-s + (−0.340 + 0.340i)22-s + (0.248 + 0.248i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $0.481 - 0.876i$
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: \((2,\ 1071,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033113069\)
\(L(\frac12)\) \(\approx\) \(2.033113069\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.82 + 1.52i)T \)
good2 \( 1 - 0.569iT - 2T^{2} \)
5 \( 1 + (0.119 - 0.119i)T - 5iT^{2} \)
11 \( 1 + (-2.80 - 2.80i)T + 11iT^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
19 \( 1 - 2.80iT - 19T^{2} \)
23 \( 1 + (-1.19 - 1.19i)T + 23iT^{2} \)
29 \( 1 + (-3.57 + 3.57i)T - 29iT^{2} \)
31 \( 1 + (-6.60 + 6.60i)T - 31iT^{2} \)
37 \( 1 + (3.66 - 3.66i)T - 37iT^{2} \)
41 \( 1 + (-1.15 - 1.15i)T + 41iT^{2} \)
43 \( 1 - 8.18iT - 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + 4.82iT - 53T^{2} \)
59 \( 1 - 4.45iT - 59T^{2} \)
61 \( 1 + (-2.83 - 2.83i)T + 61iT^{2} \)
67 \( 1 + 7.86T + 67T^{2} \)
71 \( 1 + (-8.00 + 8.00i)T - 71iT^{2} \)
73 \( 1 + (-7.42 + 7.42i)T - 73iT^{2} \)
79 \( 1 + (-2.56 - 2.56i)T + 79iT^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 + (12.1 - 12.1i)T - 97iT^{2} \)
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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

−9.986415255769928750547672911328, −9.245107895460937132826350008803, −8.183397351690539962912920340213, −7.42864827374733884244237678094, −6.60094098312719743092705232955, −6.13385267260710200299947213145, −4.86512394617347879071178807180, −3.84966701529995225666179084907, −2.65426717518296522439684790281, −1.45862816800188998804548955661, 1.00193922170488387709264725917, 2.35756986026825329310534299045, 3.29901041906757037145239249825, 4.26680703133306759978928496612, 5.60888476037866550754153449531, 6.62058576309460030830717504273, 6.84703006710982551370056695284, 8.380066149258470357767412535473, 8.803831460100648610019018044579, 9.881318226191844991158858025143

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