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Properties

Label 2-73-73.49-c1-0-0
Degree $2$
Conductor $73$
Sign $0.892 - 0.451i$
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  + (−0.233 − 0.405i)2-s + 2.48i·3-s + (0.890 − 1.54i)4-s + (−0.733 + 0.196i)5-s + (1.00 − 0.580i)6-s + (3.06 + 3.06i)7-s − 1.76·8-s − 3.16·9-s + (0.251 + 0.251i)10-s + (−1.54 − 5.75i)11-s + (3.83 + 2.21i)12-s + (−4.06 − 1.08i)13-s + (0.525 − 1.96i)14-s + (−0.488 − 1.82i)15-s + (−1.36 − 2.36i)16-s + (0.444 + 0.444i)17-s + ⋯
L(s)  = 1  + (−0.165 − 0.286i)2-s + 1.43i·3-s + (0.445 − 0.771i)4-s + (−0.327 + 0.0878i)5-s + (0.410 − 0.237i)6-s + (1.16 + 1.16i)7-s − 0.625·8-s − 1.05·9-s + (0.0794 + 0.0794i)10-s + (−0.464 − 1.73i)11-s + (1.10 + 0.638i)12-s + (−1.12 − 0.302i)13-s + (0.140 − 0.524i)14-s + (−0.126 − 0.470i)15-s + (−0.341 − 0.592i)16-s + (0.107 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.892 - 0.451i$
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: \((2,\ 73,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889963 + 0.212353i\)
\(L(\frac12)\) \(\approx\) \(0.889963 + 0.212353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (5.44 + 6.58i)T \)
good2 \( 1 + (0.233 + 0.405i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 2.48iT - 3T^{2} \)
5 \( 1 + (0.733 - 0.196i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.06 - 3.06i)T + 7iT^{2} \)
11 \( 1 + (1.54 + 5.75i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.06 + 1.08i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.444 - 0.444i)T + 17iT^{2} \)
19 \( 1 + (-2.34 + 1.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.49 + 1.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.16 - 1.38i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.60 - 0.965i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.97 + 5.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.90 - 6.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.273 + 0.273i)T - 43iT^{2} \)
47 \( 1 + (0.269 + 1.00i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.31 - 4.92i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.580 + 2.16i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.64 + 0.951i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.64 - 3.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.93 - 8.55i)T + (-35.5 + 61.4i)T^{2} \)
79 \( 1 + (-6.05 + 3.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.62 - 2.62i)T + 83iT^{2} \)
89 \( 1 + (-4.59 - 7.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.49iT - 97T^{2} \)
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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

−14.95210447789914625293046306532, −14.11372185239140488530873481005, −11.91390143497997131329421468039, −11.22614156743327897661273644082, −10.35270281332219080944239339194, −9.241778101839965108347092532172, −8.131543993585052365921927455622, −5.76976822691884650479611847478, −4.93895829032383341207318619357, −2.86374836602940847752107260282, 2.06442456773439672978256101160, 4.53251634700283680033938039650, 6.82047033101921528855184868340, 7.62953765732618892910896097259, 7.938727460530050443999290805392, 10.08840749648046964271538443076, 11.84402199761686723657726047954, 12.12835489446472722692326498445, 13.36379188306938052020043126762, 14.39689154921012764268088422453

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