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Properties

Label 2-73-73.70-c1-0-0
Degree $2$
Conductor $73$
Sign $-0.954 - 0.298i$
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  + (−1.21 + 2.10i)2-s + 0.455i·3-s + (−1.96 − 3.40i)4-s + (−0.913 + 3.40i)5-s + (−0.960 − 0.554i)6-s + (−1.73 − 1.73i)7-s + 4.70·8-s + 2.79·9-s + (−6.07 − 6.07i)10-s + (−0.546 − 0.146i)11-s + (1.55 − 0.895i)12-s + (1.24 + 4.64i)13-s + (5.75 − 1.54i)14-s + (−1.55 − 0.416i)15-s + (−1.80 + 3.12i)16-s + (0.438 + 0.438i)17-s + ⋯
L(s)  = 1  + (−0.861 + 1.49i)2-s + 0.262i·3-s + (−0.983 − 1.70i)4-s + (−0.408 + 1.52i)5-s + (−0.392 − 0.226i)6-s + (−0.654 − 0.654i)7-s + 1.66·8-s + 0.930·9-s + (−1.92 − 1.92i)10-s + (−0.164 − 0.0441i)11-s + (0.447 − 0.258i)12-s + (0.345 + 1.28i)13-s + (1.53 − 0.412i)14-s + (−0.400 − 0.107i)15-s + (−0.450 + 0.780i)16-s + (0.106 + 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(0.582907\)
Root analytic conductor: \(0.763484\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 73,\ (\ :1/2),\ -0.954 - 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0820253 + 0.537962i\)
\(L(\frac12)\) \(\approx\) \(0.0820253 + 0.537962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (-5.41 - 6.60i)T \)
good2 \( 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 0.455iT - 3T^{2} \)
5 \( 1 + (0.913 - 3.40i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.73 + 1.73i)T + 7iT^{2} \)
11 \( 1 + (0.546 + 0.146i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.24 - 4.64i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.438 - 0.438i)T + 17iT^{2} \)
19 \( 1 + (-6.38 - 3.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.907 + 0.523i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.05 + 7.66i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (2.36 + 8.81i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.197 + 0.341i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.24 - 5.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.76 + 3.76i)T - 43iT^{2} \)
47 \( 1 + (5.45 + 1.46i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.26 + 0.607i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.57 + 1.22i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.49 - 4.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.16 + 0.670i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.83 - 3.17i)T + (-35.5 - 61.4i)T^{2} \)
79 \( 1 + (-4.03 - 2.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.62 + 2.62i)T + 83iT^{2} \)
89 \( 1 + (-0.833 + 1.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.67iT - 97T^{2} \)
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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

−15.33405164778708156725526104462, −14.45023705576139774987231552928, −13.53508459116549384404721959131, −11.43791861452305383175558005768, −10.12327891599769463857442165083, −9.555718312550569886925382459404, −7.70068345539524083297811301017, −7.08272530584898712460805059504, −6.12493473657783387351572410554, −3.90981186437798635349402819085, 1.10414595671770536804622163242, 3.28622707858785150657172855753, 5.11423052020902526830215383292, 7.63809134543564860384131007699, 8.841190081872529534270060469256, 9.502052899465411083280580337435, 10.75351560547471103371305967789, 12.15494763437164555477113382752, 12.62136237481682452597151888921, 13.27559279795010791644175654355

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