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Properties

Label 1-73-73.63-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.515 - 0.857i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.515 - 0.857i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.515 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134826188 - 2.005786081i\)
\(L(\frac12)\) \(\approx\) \(1.134826188 - 2.005786081i\)
\(L(1)\) \(\approx\) \(1.358769777 - 0.8900201277i\)
\(L(1)\) \(\approx\) \(1.358769777 - 0.8900201277i\)

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

−31.57102333870888070692995673749, −31.1079459845855400185712877712, −29.51018318819220991411566896810, −28.58044181298740134014483464637, −27.27570875546961634564002208940, −26.09541006089712584287882467663, −25.26613557247310976388870767243, −23.48841257961521045656791255094, −22.89456560904349500254945563493, −21.75842332671643611903463029562, −21.08501939989334485449925024327, −19.6811164356126362457435583389, −18.61469107442231963957275233959, −16.23581745858481225717965939448, −15.96074936658044587556302809412, −14.7917791306383213176082048546, −13.7514086710980095559296561217, −12.05188364064032253244312129708, −11.17430609544537868724370038783, −9.98375901660653541452765718231, −8.19171666947260296135734750219, −6.44942824949315610164168113121, −5.28541283261449928728674820177, −3.669965251251715090088802596663, −2.93219061627646103100751987354, 0.856474701081668445318864478202, 2.84160756697248414332860142536, 4.35468296169592235276264791210, 5.911997852130783030805103277401, 7.21152376905639316408500033935, 8.19128407813502132675430507252, 10.470598832761206126037611106019, 11.88481268790484595171095976888, 12.90724027462041983484612095732, 13.38810523086623210986374242853, 15.029635684729772817623556974915, 16.16455552831355132988236209668, 17.32464809219824496715357612232, 19.064732561034343599390235454732, 20.0298570199261837359890712490, 20.74544081904391540716889606674, 22.66595630789748244204083157953, 23.344481105230170000737844086356, 24.02495097937887589958745760333, 25.23860518695063511063845875604, 26.13920408801505860662655190994, 28.23538900520646511320710469845, 28.877765266770133996612796534673, 30.28485317152252446130104133566, 30.6915940875182680049003277638

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