Properties

Label 2-71-71.5-c1-0-3
Degree $2$
Conductor $71$
Sign $0.269 + 0.963i$
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  + (−0.300 − 0.924i)2-s + (−0.143 − 0.441i)3-s + (0.853 − 0.620i)4-s + (−1.37 − 1.00i)5-s + (−0.364 + 0.265i)6-s + (0.656 + 2.02i)7-s + (−2.40 − 1.74i)8-s + (2.25 − 1.63i)9-s + (−0.511 + 1.57i)10-s + (0.603 + 1.85i)11-s + (−0.396 − 0.287i)12-s + (−1.04 + 3.22i)13-s + (1.67 − 1.21i)14-s + (−0.244 + 0.751i)15-s + (−0.239 + 0.737i)16-s + (−1.34 + 4.14i)17-s + ⋯
L(s)  = 1  + (−0.212 − 0.653i)2-s + (−0.0828 − 0.254i)3-s + (0.426 − 0.310i)4-s + (−0.615 − 0.447i)5-s + (−0.148 + 0.108i)6-s + (0.248 + 0.763i)7-s + (−0.849 − 0.617i)8-s + (0.750 − 0.545i)9-s + (−0.161 + 0.497i)10-s + (0.181 + 0.559i)11-s + (−0.114 − 0.0831i)12-s + (−0.290 + 0.895i)13-s + (0.446 − 0.324i)14-s + (−0.0630 + 0.193i)15-s + (−0.0598 + 0.184i)16-s + (−0.326 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.269 + 0.963i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ 0.269 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683353 - 0.518528i\)
\(L(\frac12)\) \(\approx\) \(0.683353 - 0.518528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-1.58 - 8.27i)T \)
good2 \( 1 + (0.300 + 0.924i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.143 + 0.441i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.37 + 1.00i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.656 - 2.02i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.603 - 1.85i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.04 - 3.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.34 - 4.14i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.31 - 4.05i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.94T + 23T^{2} \)
29 \( 1 + (5.88 + 4.27i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.17 + 6.69i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + 5.44T + 37T^{2} \)
41 \( 1 + 0.255T + 41T^{2} \)
43 \( 1 + (-6.62 - 4.81i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.0385 - 0.118i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.79 + 4.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.81 - 3.49i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.63 - 5.01i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.38 - 5.36i)T + (20.7 - 63.7i)T^{2} \)
73 \( 1 + (3.67 + 11.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.68 + 5.58i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.980 + 0.712i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-9.72 - 7.06i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−14.79589246258652917708290654353, −12.85329320661811023435592833626, −12.11502523939337549822251035283, −11.40182960082516394257147273949, −9.956120047469464476361634068541, −8.969624713610810116022521842427, −7.37396243199820040050743525470, −6.03724286460370738246532622010, −4.09347890885586106255255950687, −1.84391671828380018496428100952, 3.28390981502080037068037936971, 5.13483579359288093017639544557, 7.13657488506346799705053222065, 7.44760176713440503056541768186, 8.990139144982418446273966933096, 10.71722965250968327572974871353, 11.29419827831246390999069327950, 12.76350816380933548495388765917, 14.04029941234804878852875241184, 15.29213844014059492361612242526

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