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Properties

Label 2-73-73.49-c1-0-4
Degree $2$
Conductor $73$
Sign $-0.974 + 0.225i$
Analytic cond. $0.582907$
Root an. cond. $0.763484$
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  + (−1.24 − 2.15i)2-s − 2.90i·3-s + (−2.10 + 3.64i)4-s + (1.39 − 0.372i)5-s + (−6.27 + 3.62i)6-s + (2.87 + 2.87i)7-s + 5.50·8-s − 5.45·9-s + (−2.53 − 2.53i)10-s + (−0.224 − 0.838i)11-s + (10.5 + 6.11i)12-s + (−2.18 − 0.585i)13-s + (2.61 − 9.77i)14-s + (−1.08 − 4.04i)15-s + (−2.64 − 4.58i)16-s + (0.00830 + 0.00830i)17-s + ⋯
L(s)  = 1  + (−0.880 − 1.52i)2-s − 1.67i·3-s + (−1.05 + 1.82i)4-s + (0.621 − 0.166i)5-s + (−2.56 + 1.47i)6-s + (1.08 + 1.08i)7-s + 1.94·8-s − 1.81·9-s + (−0.802 − 0.802i)10-s + (−0.0677 − 0.252i)11-s + (3.05 + 1.76i)12-s + (−0.606 − 0.162i)13-s + (0.699 − 2.61i)14-s + (−0.279 − 1.04i)15-s + (−0.661 − 1.14i)16-s + (0.00201 + 0.00201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(0.582907\)
Root analytic conductor: \(0.763484\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 73,\ (\ :1/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0753670 - 0.659149i\)
\(L(\frac12)\) \(\approx\) \(0.0753670 - 0.659149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (-3.72 - 7.69i)T \)
good2 \( 1 + (1.24 + 2.15i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 2.90iT - 3T^{2} \)
5 \( 1 + (-1.39 + 0.372i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.87 - 2.87i)T + 7iT^{2} \)
11 \( 1 + (0.224 + 0.838i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.18 + 0.585i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.00830 - 0.00830i)T + 17iT^{2} \)
19 \( 1 + (0.315 - 0.182i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.47 - 3.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (9.01 + 2.41i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.51 - 1.20i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.95 + 5.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.95 - 3.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.903 - 0.903i)T - 43iT^{2} \)
47 \( 1 + (-0.796 - 2.97i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.926 + 3.45i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.862 - 3.21i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.60 - 4.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.27 + 1.31i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.58 + 7.94i)T + (-35.5 + 61.4i)T^{2} \)
79 \( 1 + (6.89 - 3.98i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.71 + 6.71i)T + 83iT^{2} \)
89 \( 1 + (1.94 + 3.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.75iT - 97T^{2} \)
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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

−13.43757350868181627010926629066, −12.76711140971275114507841669472, −11.77850418111334534019757494063, −11.21370917989314348026690745400, −9.483417889318457553030216516075, −8.505224504025460084312869121797, −7.55204957149168873342011403154, −5.57711400710902234437604836021, −2.55441312245444866236410383617, −1.54927322932926526713938602307, 4.47087622693690998193447837696, 5.39173905363126725915805802928, 7.03308648322594174497347143681, 8.318363496651319295244071900164, 9.466436160264947561179555067956, 10.19780199657250254927975854663, 11.08019460178358312634549032190, 13.75911595530078833064185006143, 14.70263533050121792222872162929, 15.07695202825154169203237971952

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