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Properties

Label 1-73-73.13-r1-0-0
Degree $1$
Conductor $73$
Sign $0.351 - 0.936i$
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  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (0.422 − 0.906i)5-s + (−0.984 + 0.173i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.906 + 0.422i)11-s + (0.984 + 0.173i)12-s + (−0.573 + 0.819i)13-s + (−0.819 − 0.573i)14-s + (−0.0871 − 0.996i)15-s + (0.173 + 0.984i)16-s + (0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (0.422 − 0.906i)5-s + (−0.984 + 0.173i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.906 + 0.422i)11-s + (0.984 + 0.173i)12-s + (−0.573 + 0.819i)13-s + (−0.819 − 0.573i)14-s + (−0.0871 − 0.996i)15-s + (0.173 + 0.984i)16-s + (0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.351 - 0.936i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.351 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425367024 - 0.9875028322i\)
\(L(\frac12)\) \(\approx\) \(1.425367024 - 0.9875028322i\)
\(L(1)\) \(\approx\) \(1.083279429 - 0.4570187557i\)
\(L(1)\) \(\approx\) \(1.083279429 - 0.4570187557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.422 - 0.906i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (0.906 + 0.422i)T \)
13 \( 1 + (-0.573 + 0.819i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.422 - 0.906i)T \)
31 \( 1 + (0.996 + 0.0871i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (-0.819 + 0.573i)T \)
53 \( 1 + (-0.906 + 0.422i)T \)
59 \( 1 + (0.819 + 0.573i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.984 + 0.173i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−31.56562734838240741943581528990, −30.10658005580949174927557689479, −29.5672793252166242898426473921, −27.50602957139325148744180810715, −27.26442033122991328457298069776, −26.19986348757330356822721593712, −25.1285959370039980303590317798, −24.48023510727040196373153362678, −22.65433823823711743264099381853, −21.30896728393809501816124089055, −20.29954783136438322311593885178, −19.24034547742558485259912483345, −18.142444942604919352210780515824, −17.0962118050859920797955284445, −15.70291228667500720635735384212, −14.51976434333654318696575260516, −14.07946946847844310907699695428, −11.52651795422581231225259596080, −10.37387687154763554443985582536, −9.47985983319239563716040230050, −8.1005205868093977788195008616, −7.15985036159791771576426830905, −5.38865057156389592349833945795, −3.23993906531667224685425004395, −1.706786337311872731698140083648, 1.28142595831039621197569852914, 2.235895061848301567531501512109, 4.31644603103522179349376282887, 6.59098731241537183700810496922, 8.02115201233791221386709570331, 8.87224084140084745758125318822, 9.80203198095480286735455187804, 11.713998195877929341873121998382, 12.55797488445786286551526456804, 14.05316268221855781149282456715, 15.3085732876231702355153674479, 16.979934051057990750617599229171, 17.68086601879281613738805542990, 18.99499100224650643240945656635, 19.94755600547551414512273607796, 20.837578783753473298525581865797, 21.70294729940462984709139008660, 24.25982608728207761662151606098, 24.52560100200045023520285353582, 25.70232658320996899770806199543, 26.71358988921551360903296802022, 27.93388918209066770860632409532, 28.738449657905506314333201538681, 30.12472060128160864190693561375, 30.6973475799232561946320629897

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