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Properties

Label 2-73-73.70-c1-0-4
Degree $2$
Conductor $73$
Sign $0.166 + 0.985i$
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  + (0.593 − 1.02i)2-s − 3.06i·3-s + (0.295 + 0.512i)4-s + (−1.03 + 3.85i)5-s + (−3.14 − 1.81i)6-s + (−1.17 − 1.17i)7-s + 3.07·8-s − 6.38·9-s + (3.34 + 3.34i)10-s + (0.745 + 0.199i)11-s + (1.56 − 0.906i)12-s + (0.0599 + 0.223i)13-s + (−1.89 + 0.508i)14-s + (11.8 + 3.16i)15-s + (1.23 − 2.13i)16-s + (1.58 + 1.58i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.726i)2-s − 1.76i·3-s + (0.147 + 0.256i)4-s + (−0.462 + 1.72i)5-s + (−1.28 − 0.742i)6-s + (−0.442 − 0.442i)7-s + 1.08·8-s − 2.12·9-s + (1.05 + 1.05i)10-s + (0.224 + 0.0602i)11-s + (0.453 − 0.261i)12-s + (0.0166 + 0.0620i)13-s + (−0.507 + 0.135i)14-s + (3.05 + 0.817i)15-s + (0.308 − 0.534i)16-s + (0.385 + 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.166 + 0.985i$
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: \((2,\ 73,\ (\ :1/2),\ 0.166 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834958 - 0.705509i\)
\(L(\frac12)\) \(\approx\) \(0.834958 - 0.705509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (-3.43 + 7.82i)T \)
good2 \( 1 + (-0.593 + 1.02i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 3.06iT - 3T^{2} \)
5 \( 1 + (1.03 - 3.85i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.17 + 1.17i)T + 7iT^{2} \)
11 \( 1 + (-0.745 - 0.199i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.0599 - 0.223i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.58 - 1.58i)T + 17iT^{2} \)
19 \( 1 + (5.13 + 2.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.739 + 0.427i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.958 - 3.57i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.110 + 0.412i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.311 + 0.539i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.104 + 0.180i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0314 - 0.0314i)T - 43iT^{2} \)
47 \( 1 + (7.11 + 1.90i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-11.9 + 3.20i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (13.5 - 3.63i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.86 + 3.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.34 - 4.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.13 + 5.43i)T + (-35.5 - 61.4i)T^{2} \)
79 \( 1 + (-2.82 - 1.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.07 - 3.07i)T + 83iT^{2} \)
89 \( 1 + (6.67 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.46iT - 97T^{2} \)
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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

−13.98806065362018007127209061105, −13.15014392400445514347497296529, −12.23492716188207776503530390692, −11.31726549175578434072293427409, −10.52419797978468528014019715817, −8.096449428531745608723086580429, −7.05042941298766025851775102862, −6.56622349794200986882636891753, −3.50665647079416143782249614317, −2.31399372535348298443778950223, 4.06292218582996448000299119271, 4.96172738445381631933735005652, 5.91135568575729453410915147784, 8.205270279315201471883020877645, 9.205081662991657222321607196284, 10.16737543689389730741327738671, 11.51416376872216185295686594681, 12.77039199412076400216415657754, 14.18385105111779533835303491671, 15.31416644476799933568329212030

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