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Properties

Label 1-73-73.30-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.512 + 0.858i$
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  + (−0.5 − 0.866i)2-s + i·3-s + (−0.5 + 0.866i)4-s + (0.258 + 0.965i)5-s + (0.866 − 0.5i)6-s + (0.707 − 0.707i)7-s + 8-s − 9-s + (0.707 − 0.707i)10-s + (−0.965 + 0.258i)11-s + (−0.866 − 0.5i)12-s + (−0.258 + 0.965i)13-s + (−0.965 − 0.258i)14-s + (−0.965 + 0.258i)15-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + i·3-s + (−0.5 + 0.866i)4-s + (0.258 + 0.965i)5-s + (0.866 − 0.5i)6-s + (0.707 − 0.707i)7-s + 8-s − 9-s + (0.707 − 0.707i)10-s + (−0.965 + 0.258i)11-s + (−0.866 − 0.5i)12-s + (−0.258 + 0.965i)13-s + (−0.965 − 0.258i)14-s + (−0.965 + 0.258i)15-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.512 + 0.858i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3892432809 + 0.6855233304i\)
\(L(\frac12)\) \(\approx\) \(0.3892432809 + 0.6855233304i\)
\(L(1)\) \(\approx\) \(0.7075234842 + 0.2005262086i\)
\(L(1)\) \(\approx\) \(0.7075234842 + 0.2005262086i\)

Euler product

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

−31.37012222039928366118189005615, −29.60034748448028422067176354304, −28.5690221797165019081988216479, −27.81871559900207776916046544284, −26.40824765782159792976925650609, −25.02555814598568487412858089598, −24.628521722245437801573212399682, −23.79867332770811202260121438882, −22.52824602984010133316206591473, −20.73236300425022403165140334966, −19.62405387961856794476797972481, −18.10734349846782821789675767760, −17.88406770024013910837287979581, −16.44440937044497069898787249459, −15.27895478005581722764032774653, −13.847535723061704754808781331167, −12.9304531484246575164423549975, −11.49013709520727389215793916189, −9.624476419277536456325637007506, −8.280000208145340912738284369643, −7.72170295151269082331373714929, −5.84763494813521909938061250377, −5.15251778855516215403371881687, −2.05482847074268775120382281156, −0.44840107665740582767109459563, 2.1862665024232489047302050308, 3.64306234718401689743042024721, 4.90796943834347003214459910755, 7.17707022927555770926266550866, 8.64260489797638113261800040812, 10.07948568643609063516472362355, 10.68558623594899950791065820372, 11.67044636402630419260180700797, 13.59828089308889765875343673771, 14.58597209995947298144899904844, 16.10562096973504219894471840208, 17.40979641079207730210146599006, 18.21791563744838361050080039493, 19.68712603143023720612517685039, 20.689050121547737992170061578535, 21.62966545295421572540445563650, 22.39573332068608896584325992371, 23.74156909584748106903294264300, 25.9740296294229159244988747026, 26.373269077156719372065836927521, 27.20450728237699843103958435538, 28.44756902478635067965993161304, 29.28519948401801166098386145399, 30.68235670691916654166668553602, 31.19525180274646856723539273721

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