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Properties

Label 2-73-73.49-c1-0-3
Degree $2$
Conductor $73$
Sign $0.585 + 0.810i$
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  + (0.0339 + 0.0587i)2-s − 1.73i·3-s + (0.997 − 1.72i)4-s + (−2.59 + 0.694i)5-s + (0.102 − 0.0589i)6-s + (0.823 + 0.823i)7-s + 0.271·8-s − 0.0141·9-s + (−0.128 − 0.128i)10-s + (1.27 + 4.74i)11-s + (−3.00 − 1.73i)12-s + (4.09 + 1.09i)13-s + (−0.0204 + 0.0763i)14-s + (1.20 + 4.50i)15-s + (−1.98 − 3.44i)16-s + (−3.48 − 3.48i)17-s + ⋯
L(s)  = 1  + (0.0240 + 0.0415i)2-s − 1.00i·3-s + (0.498 − 0.864i)4-s + (−1.15 + 0.310i)5-s + (0.0416 − 0.0240i)6-s + (0.311 + 0.311i)7-s + 0.0958·8-s − 0.00471·9-s + (−0.0407 − 0.0407i)10-s + (0.383 + 1.43i)11-s + (−0.866 − 0.500i)12-s + (1.13 + 0.304i)13-s + (−0.00547 + 0.0204i)14-s + (0.311 + 1.16i)15-s + (−0.496 − 0.860i)16-s + (−0.844 − 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.585 + 0.810i$
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: \((2,\ 73,\ (\ :1/2),\ 0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829050 - 0.423928i\)
\(L(\frac12)\) \(\approx\) \(0.829050 - 0.423928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (-5.64 + 6.41i)T \)
good2 \( 1 + (-0.0339 - 0.0587i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 1.73iT - 3T^{2} \)
5 \( 1 + (2.59 - 0.694i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.823 - 0.823i)T + 7iT^{2} \)
11 \( 1 + (-1.27 - 4.74i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.09 - 1.09i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.48 + 3.48i)T + 17iT^{2} \)
19 \( 1 + (6.00 - 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.846 + 0.488i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.74 - 1.27i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.234 + 0.0628i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.324 + 0.561i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.76 - 6.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.45 - 3.45i)T - 43iT^{2} \)
47 \( 1 + (2.08 + 7.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.988 + 3.69i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.75 - 6.55i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.03 - 5.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.32 - 4.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 4.16i)T + (-35.5 + 61.4i)T^{2} \)
79 \( 1 + (-8.33 + 4.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.35 + 5.35i)T + 83iT^{2} \)
89 \( 1 + (6.08 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.81iT - 97T^{2} \)
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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

−14.68109142918181793145337124570, −13.32635047368381651213093997737, −12.06613379092859998533493635480, −11.43102362836776395336306251085, −10.14763660513046583268887261168, −8.437475482424325283139506640359, −7.14163941088495444236895547765, −6.46494374537178182055679895742, −4.42814833544235327636191000279, −1.88371965047175996034022647625, 3.59438477809704294661121643248, 4.30274666185620035084912014953, 6.51121314796509447051745143941, 8.191043429613288959847488705968, 8.715430356999903340654083456993, 10.92557761393996657375326017154, 11.06663358151939193926408460048, 12.48292168807280573747573885790, 13.61277837054347220095450592762, 15.32983866350794880015282528708

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