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Properties

Label 1-73-73.26-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.357 - 0.933i$
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 + (0.906 − 0.422i)5-s + (0.984 + 0.173i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.422 + 0.906i)11-s + (−0.984 + 0.173i)12-s + (0.819 − 0.573i)13-s + (0.573 + 0.819i)14-s + (−0.996 − 0.0871i)15-s + (0.173 − 0.984i)16-s + (−0.965 − 0.258i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.866 − 0.5i)3-s + (0.766 − 0.642i)4-s + (0.906 − 0.422i)5-s + (0.984 + 0.173i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.422 + 0.906i)11-s + (−0.984 + 0.173i)12-s + (0.819 − 0.573i)13-s + (0.573 + 0.819i)14-s + (−0.996 − 0.0871i)15-s + (0.173 − 0.984i)16-s + (−0.965 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.357 - 0.933i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.357 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4088131940 - 0.5944929342i\)
\(L(\frac12)\) \(\approx\) \(0.4088131940 - 0.5944929342i\)
\(L(1)\) \(\approx\) \(0.5860461604 - 0.2055288782i\)
\(L(1)\) \(\approx\) \(0.5860461604 - 0.2055288782i\)

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 \( 1 + (0.906 - 0.422i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.422 + 0.906i)T \)
13 \( 1 + (0.819 - 0.573i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.906 - 0.422i)T \)
31 \( 1 + (0.0871 + 0.996i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (0.573 - 0.819i)T \)
53 \( 1 + (-0.422 + 0.906i)T \)
59 \( 1 + (-0.573 - 0.819i)T \)
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

−31.57191029890717237617300544714, −30.00176654078064440096863785853, −29.16481672288147227077896516673, −28.45673915079036273362827038158, −27.44981568603207770834068111807, −26.35904755729454456570927871821, −25.399324691267808426632310743646, −24.21027816972620602619511648895, −22.374288894233694604066678222690, −21.6488766072831364648429764165, −20.87186463760591661067316411165, −18.99804931833626895840590124043, −18.291680941712052287898857307129, −17.18962005265630229039013170672, −16.269936501638585815840409706414, −15.09625990294892670473377574128, −13.15526667799569628975644178394, −11.67473913471045111861226095596, −10.88967448087434980476097969149, −9.62418484394379760874340667569, −8.7688810320906765678133923069, −6.562997859816718284588525819113, −5.83066247432272865300861590236, −3.49180301974856514454791413234, −1.6777125819572448823540567455, 0.549264606168342424805172506571, 1.89496747052554594074664705706, 4.90634637996989307433660649488, 6.37514989304405470922525578019, 7.124293498468047932264833944285, 8.81808543245663019474301778155, 10.17889949866940543059863148813, 11.014442319974818077017917520069, 12.67068671730824718927713680797, 13.79366325666443081700476013954, 15.609330322819505242133837326020, 16.86124315681510871252371173090, 17.45344774009839922863001094419, 18.28094764124006454937954195281, 19.77687291788514692587756366228, 20.70619182830550903659272402471, 22.42938193554447595077877731057, 23.51138326152609032306899857327, 24.59625304196595206828309824408, 25.44585609359144039037757174497, 26.59796941987309680208646862755, 28.03521814347594803616570002546, 28.56040793714143960227566538849, 29.65367059755281371171786773817, 30.36376259034764249573332165951

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