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Properties

Label 2-73-73.49-c1-0-2
Degree $2$
Conductor $73$
Sign $0.271 + 0.962i$
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  + (−0.723 − 1.25i)2-s + 0.458i·3-s + (−0.0481 + 0.0833i)4-s + (2.03 − 0.546i)5-s + (0.575 − 0.332i)6-s + (−2.38 − 2.38i)7-s − 2.75·8-s + 2.78·9-s + (−2.16 − 2.16i)10-s + (1.13 + 4.24i)11-s + (−0.0382 − 0.0220i)12-s + (−2.85 − 0.764i)13-s + (−1.26 + 4.71i)14-s + (0.250 + 0.934i)15-s + (2.09 + 3.62i)16-s + (5.36 + 5.36i)17-s + ⋯
L(s)  = 1  + (−0.511 − 0.886i)2-s + 0.264i·3-s + (−0.0240 + 0.0416i)4-s + (0.911 − 0.244i)5-s + (0.234 − 0.135i)6-s + (−0.901 − 0.901i)7-s − 0.974·8-s + 0.929·9-s + (−0.683 − 0.683i)10-s + (0.343 + 1.28i)11-s + (−0.0110 − 0.00637i)12-s + (−0.791 − 0.212i)13-s + (−0.337 + 1.26i)14-s + (0.0646 + 0.241i)15-s + (0.522 + 0.905i)16-s + (1.30 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.271 + 0.962i$
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: \((2,\ 73,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640832 - 0.484894i\)
\(L(\frac12)\) \(\approx\) \(0.640832 - 0.484894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (-7.50 + 4.08i)T \)
good2 \( 1 + (0.723 + 1.25i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 0.458iT - 3T^{2} \)
5 \( 1 + (-2.03 + 0.546i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.38 + 2.38i)T + 7iT^{2} \)
11 \( 1 + (-1.13 - 4.24i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.85 + 0.764i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.36 - 5.36i)T + 17iT^{2} \)
19 \( 1 + (2.10 - 1.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.13 + 1.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.66 + 1.51i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.429 - 0.115i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.93 + 5.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.80 + 3.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 4.83i)T - 43iT^{2} \)
47 \( 1 + (0.780 + 2.91i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.89 - 10.8i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.40 + 8.98i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.21 + 4.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.97 - 5.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.54 - 6.13i)T + (-35.5 + 61.4i)T^{2} \)
79 \( 1 + (8.47 - 4.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.55 + 5.55i)T + 83iT^{2} \)
89 \( 1 + (3.64 + 6.31i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−14.40896850725847525248277988175, −12.82666022490105296975924595165, −12.42025409105180027053967567352, −10.50941238324766504235784058256, −9.971101567156987052155028438980, −9.450592087550131163758424749953, −7.35103327013006252889971675245, −5.93505752885343516028859433707, −3.95037004690564142640566508105, −1.80730583519354466387419607056, 2.89957091277877891348150661013, 5.71516605126907604042485700927, 6.53920982919477209328886666882, 7.72479112709902728228648406440, 9.237609165995347138690453234151, 9.792625455208315131724407588917, 11.72748664310337572884322639370, 12.72373967090626807589677165357, 13.88884495088970000227294373425, 15.02324289683127441154537683576

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