Properties

Label 2-73-73.70-c1-0-1
Degree $2$
Conductor $73$
Sign $0.819 - 0.573i$
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  + (−0.202 + 0.350i)2-s − 0.0458i·3-s + (0.917 + 1.58i)4-s + (0.0766 − 0.286i)5-s + (0.0160 + 0.00928i)6-s + (0.00171 + 0.00171i)7-s − 1.55·8-s + 2.99·9-s + (0.0848 + 0.0848i)10-s + (−3.03 − 0.813i)11-s + (0.0729 − 0.0421i)12-s + (−0.799 − 2.98i)13-s + (−0.000947 + 0.000253i)14-s + (−0.0131 − 0.00351i)15-s + (−1.52 + 2.63i)16-s + (−4.10 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.143 + 0.248i)2-s − 0.0264i·3-s + (0.458 + 0.794i)4-s + (0.0342 − 0.127i)5-s + (0.00656 + 0.00379i)6-s + (0.000646 + 0.000646i)7-s − 0.549·8-s + 0.999·9-s + (0.0268 + 0.0268i)10-s + (−0.915 − 0.245i)11-s + (0.0210 − 0.0121i)12-s + (−0.221 − 0.827i)13-s + (−0.000253 + 6.78e−5i)14-s + (−0.00338 − 0.000908i)15-s + (−0.380 + 0.658i)16-s + (−0.995 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.819 - 0.573i$
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: \((2,\ 73,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885503 + 0.279074i\)
\(L(\frac12)\) \(\approx\) \(0.885503 + 0.279074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (8.54 - 0.245i)T \)
good2 \( 1 + (0.202 - 0.350i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.0458iT - 3T^{2} \)
5 \( 1 + (-0.0766 + 0.286i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.00171 - 0.00171i)T + 7iT^{2} \)
11 \( 1 + (3.03 + 0.813i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.799 + 2.98i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.10 + 4.10i)T + 17iT^{2} \)
19 \( 1 + (2.16 + 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.40 + 0.813i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.129 + 0.483i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.37 - 5.13i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.56 - 4.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.48 - 6.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.55 - 3.55i)T - 43iT^{2} \)
47 \( 1 + (0.430 + 0.115i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.36 + 1.17i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (8.29 - 2.22i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.08 + 3.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.01 + 5.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.21 - 12.5i)T + (-35.5 - 61.4i)T^{2} \)
79 \( 1 + (-7.02 - 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.97 - 2.97i)T + 83iT^{2} \)
89 \( 1 + (-3.78 + 6.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.25iT - 97T^{2} \)
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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

−15.08678371581629773495316419952, −13.25409528197398088836610181519, −12.78944257898999819981751912490, −11.44093392517976876862571410116, −10.30545410913289068933036452693, −8.822171998713283276229289852052, −7.66677372245933987463866361499, −6.67512775960394561833258482645, −4.80360513681078546836717064781, −2.85938124396687415927740767820, 2.11347276926257703104396827397, 4.50647084234277519292556653610, 6.17131293442517796760325875810, 7.31969323566633786677255560950, 9.044585250300658395050384593495, 10.25856618060113264268647267983, 10.89942187160837549189203875252, 12.30059526688830333659377608690, 13.36077383814855775239427636127, 14.73534012136085947595158631477

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