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Properties

Label 2-71-71.57-c1-0-0
Degree $2$
Conductor $71$
Sign $-0.927 - 0.373i$
Analytic cond. $0.566937$
Root an. cond. $0.752952$
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  + (−0.699 + 2.15i)2-s + (−0.874 + 2.69i)3-s + (−2.52 − 1.83i)4-s + (3.04 − 2.21i)5-s + (−5.17 − 3.76i)6-s + (−0.0944 + 0.290i)7-s + (2.04 − 1.48i)8-s + (−4.04 − 2.94i)9-s + (2.63 + 8.09i)10-s + (0.449 − 1.38i)11-s + (7.13 − 5.18i)12-s + (1.46 + 4.52i)13-s + (−0.559 − 0.406i)14-s + (3.28 + 10.1i)15-s + (−0.158 − 0.486i)16-s + (−1.53 − 4.73i)17-s + ⋯
L(s)  = 1  + (−0.494 + 1.52i)2-s + (−0.504 + 1.55i)3-s + (−1.26 − 0.916i)4-s + (1.36 − 0.989i)5-s + (−2.11 − 1.53i)6-s + (−0.0356 + 0.109i)7-s + (0.723 − 0.525i)8-s + (−1.34 − 0.980i)9-s + (0.831 + 2.56i)10-s + (0.135 − 0.417i)11-s + (2.06 − 1.49i)12-s + (0.407 + 1.25i)13-s + (−0.149 − 0.108i)14-s + (0.849 + 2.61i)15-s + (−0.0395 − 0.121i)16-s + (−0.372 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135236 + 0.698008i\)
\(L(\frac12)\) \(\approx\) \(0.135236 + 0.698008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (8.04 + 2.48i)T \)
good2 \( 1 + (0.699 - 2.15i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.874 - 2.69i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-3.04 + 2.21i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.0944 - 0.290i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.449 + 1.38i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.46 - 4.52i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.53 + 4.73i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.873 - 2.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.29T + 23T^{2} \)
29 \( 1 + (-2.50 + 1.81i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.352 + 1.08i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + 1.09T + 37T^{2} \)
41 \( 1 + 6.01T + 41T^{2} \)
43 \( 1 + (2.12 - 1.54i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.43 + 4.41i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.35 - 0.982i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-10.6 + 7.76i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.17 + 3.62i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (12.2 + 8.89i)T + (20.7 + 63.7i)T^{2} \)
73 \( 1 + (3.17 - 9.77i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.69 - 5.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.58 + 1.87i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.87 - 3.54i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 - 9.88T + 97T^{2} \)
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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

−15.61310717560644426538566821561, −14.37508738087296326506049984673, −13.57949869479522971941942194968, −11.64082879387100374508734621554, −10.06627078415570961425541640323, −9.257771244320265510367395647268, −8.711630427946539334512310351481, −6.48773770950302019566123991760, −5.51221797666917506108968848209, −4.63189256511743155365071950375, 1.57321639499304235469686711786, 2.80829491827303825100042367207, 5.91863316200858239147373239908, 6.97809508572585397683852669829, 8.639221814176321195664951347915, 10.22969341224325228113700864697, 10.77155969038665385007307394790, 11.96919236313848025946297769705, 13.16657444079138079350583794161, 13.30656415494946335464044795288

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