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Properties

Label 2-71-71.5-c1-0-2
Degree $2$
Conductor $71$
Sign $0.178 - 0.983i$
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  + (0.420 + 1.29i)2-s + (0.672 + 2.06i)3-s + (0.123 − 0.0894i)4-s + (−2.53 − 1.83i)5-s + (−2.39 + 1.73i)6-s + (−1.31 − 4.04i)7-s + (2.36 + 1.71i)8-s + (−1.40 + 1.01i)9-s + (1.31 − 4.04i)10-s + (0.802 + 2.47i)11-s + (0.267 + 0.194i)12-s + (−1.18 + 3.64i)13-s + (4.67 − 3.39i)14-s + (2.10 − 6.47i)15-s + (−1.13 + 3.49i)16-s + (−0.514 + 1.58i)17-s + ⋯
L(s)  = 1  + (0.297 + 0.914i)2-s + (0.388 + 1.19i)3-s + (0.0615 − 0.0447i)4-s + (−1.13 − 0.822i)5-s + (−0.976 + 0.709i)6-s + (−0.496 − 1.52i)7-s + (0.836 + 0.607i)8-s + (−0.467 + 0.339i)9-s + (0.415 − 1.27i)10-s + (0.242 + 0.744i)11-s + (0.0773 + 0.0561i)12-s + (−0.328 + 1.01i)13-s + (1.25 − 0.908i)14-s + (0.543 − 1.67i)15-s + (−0.283 + 0.873i)16-s + (−0.124 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.178 - 0.983i$
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: \((2,\ 71,\ (\ :1/2),\ 0.178 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831263 + 0.693979i\)
\(L(\frac12)\) \(\approx\) \(0.831263 + 0.693979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-2.18 + 8.13i)T \)
good2 \( 1 + (-0.420 - 1.29i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.672 - 2.06i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.53 + 1.83i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.31 + 4.04i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.802 - 2.47i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.18 - 3.64i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.514 - 1.58i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.14 + 6.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 + (-0.829 - 0.602i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.21 + 3.74i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 + 3.25T + 41T^{2} \)
43 \( 1 + (-5.06 - 3.68i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.919 - 2.83i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.74 - 1.99i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.332 - 0.241i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.569 + 1.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.81 - 5.67i)T + (20.7 - 63.7i)T^{2} \)
73 \( 1 + (-2.11 - 6.51i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.72 + 4.88i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.85 + 4.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.44 - 3.22i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−15.15093807465533286445079068419, −14.26240736187325483181728832963, −13.04894809465514028377485066942, −11.46432521009284826632102556322, −10.33797872519715197809735279452, −9.183072504030973993212059768391, −7.74947137559791255987315995643, −6.76109908928420193923792184671, −4.37890070270931349094769939931, −4.32870885784061187776382770645, 2.42762600723552739211620877981, 3.47225385501369763777576278199, 6.16270681860365133575650269198, 7.49775358452173775962606432777, 8.367777054292670960182743463285, 10.28921627511898018684439654416, 11.61863037087607044668479898873, 12.21305427064191085018209092231, 12.89467407654271474134353496858, 14.23835455435886603472729915694

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