↑

Properties

Label 2-73-73.24-c1-0-0
Degree $2$
Conductor $73$
Sign $0.934 - 0.355i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.983 − 1.70i)2-s + 3.07i·3-s + (−0.933 + 1.61i)4-s + (0.710 + 2.65i)5-s + (5.23 − 3.02i)6-s + (1.20 − 1.20i)7-s − 0.260·8-s − 6.43·9-s + (3.81 − 3.81i)10-s + (3.04 − 0.817i)11-s + (−4.96 − 2.86i)12-s + (0.961 − 3.58i)13-s + (−3.24 − 0.869i)14-s + (−8.14 + 2.18i)15-s + (2.12 + 3.67i)16-s + (−2.63 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.695 − 1.20i)2-s + 1.77i·3-s + (−0.466 + 0.808i)4-s + (0.317 + 1.18i)5-s + (2.13 − 1.23i)6-s + (0.456 − 0.456i)7-s − 0.0921·8-s − 2.14·9-s + (1.20 − 1.20i)10-s + (0.919 − 0.246i)11-s + (−1.43 − 0.827i)12-s + (0.266 − 0.994i)13-s + (−0.866 − 0.232i)14-s + (−2.10 + 0.563i)15-s + (0.530 + 0.919i)16-s + (−0.638 + 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(0.582907\)
Root analytic conductor: \(0.763484\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 73,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693097 + 0.127226i\)
\(L(\frac12)\) \(\approx\) \(0.693097 + 0.127226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (5.00 - 6.92i)T \)
good2 \( 1 + (0.983 + 1.70i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 3.07iT - 3T^{2} \)
5 \( 1 + (-0.710 - 2.65i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.20 + 1.20i)T - 7iT^{2} \)
11 \( 1 + (-3.04 + 0.817i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.961 + 3.58i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.63 - 2.63i)T - 17iT^{2} \)
19 \( 1 + (-3.56 + 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.30 + 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.660 + 2.46i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.43 + 9.08i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.10 - 8.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.308 - 0.533i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0522 - 0.0522i)T + 43iT^{2} \)
47 \( 1 + (-2.65 + 0.710i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.51 - 0.942i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.533 + 0.143i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.80 + 4.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.81 - 1.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.34 - 2.32i)T + (-35.5 + 61.4i)T^{2} \)
79 \( 1 + (1.52 - 0.881i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.354 - 0.354i)T - 83iT^{2} \)
89 \( 1 + (2.13 + 3.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.467iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.86159879220236979542399791848, −13.78965921466041749344966644765, −11.70630705827626807833472308970, −10.94640554884957180639059166073, −10.32583367745370671782415601303, −9.608893105209236480703119608536, −8.382574455309986419207651024118, −6.07595555128295940165684785706, −4.08585798811990095270120900264, −2.91965745836372899176250392663, 1.53583126528139858507137394849, 5.39393900144830471750490525227, 6.54818801019474363796598516540, 7.46380323634679305963683587947, 8.673839774954521374189302105040, 9.119024650792290617544534768734, 11.86686662581277399494878168191, 12.25380187129852617507357145878, 13.71801397992984799766870130100, 14.34839706884583336374188675224

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