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Properties

Label 1-73-73.10-r1-0-0
Degree $1$
Conductor $73$
Sign $0.911 - 0.411i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − i·3-s + 4-s + (0.707 + 0.707i)5-s − i·6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)11-s − i·12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s + (0.707 + 0.707i)5-s − i·6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)11-s − i·12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.377433143 - 0.7264739553i\)
\(L(\frac12)\) \(\approx\) \(3.377433143 - 0.7264739553i\)
\(L(1)\) \(\approx\) \(2.193297539 - 0.3496543215i\)
\(L(1)\) \(\approx\) \(2.193297539 - 0.3496543215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - iT \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + iT \)
67 \( 1 + iT \)
71 \( 1 - T \)
79 \( 1 - iT \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.6420195369807855095623249969, −30.383373314516706063694396668972, −29.32112970736686297617746234483, −28.19963931493521137501508224143, −27.13912784475757738251490511245, −25.67074804445111977479545959126, −24.78482665611963515717211281790, −23.59417362093740270740652775651, −22.36991998601692250644701970864, −21.553758154134309007862770916455, −20.42339031181154836260423013897, −20.04262698935650390420971828761, −17.29073524046397807954785243295, −16.86723011677337235353502327302, −15.29613042628456746856989350504, −14.47793834294038145055921185974, −13.32476966111447093043610289054, −11.9766207985374509542301509377, −10.66022116313707289718076491808, −9.5918865228543286644491426377, −7.81352719097941667896978033355, −5.9333670727277098798298648729, −4.81822486915966988447216889232, −3.882972504563546083100533587512, −1.85980780939847579611240131058, 1.83548573900774808508742908940, 2.8472586115949875064380489746, 5.041694032967537112778542359782, 6.32578602278321340435094671870, 7.154543381755321587162336173277, 8.92826091539318994254993156155, 11.08128146787278227059549504473, 11.79883506016001872502045583810, 13.164494813258069775733896848437, 14.197988565753134779208425840762, 14.82043537258545649768766602931, 16.68249444457654711135541064414, 17.94378652182877990948585688981, 19.00793541003022399018628782616, 20.24075760542591890198766824735, 21.79852768640902709367124830001, 22.19816300729218376516812052605, 23.74824835562454517907255970300, 24.5944787194427662361621929661, 25.24604945145069141881266764693, 26.60474027237630107866787672430, 28.5469109217440133934559538844, 29.37526133283508306145886944539, 30.28436276137580967502203847618, 30.99626815899719299382163639664

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