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Properties

Label 1-71-71.33-r1-0-0
Degree $1$
Conductor $71$
Sign $0.359 - 0.933i$
Analytic cond. $7.63000$
Root an. cond. $7.63000$
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  + (0.753 − 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 − 0.990i)4-s + (0.309 − 0.951i)5-s + (0.936 + 0.351i)6-s + (0.393 − 0.919i)7-s + (−0.550 − 0.834i)8-s + (−0.550 + 0.834i)9-s + (−0.393 − 0.919i)10-s + (0.0448 + 0.998i)11-s + (0.936 − 0.351i)12-s + (0.0448 − 0.998i)13-s + (−0.309 − 0.951i)14-s + (0.983 − 0.178i)15-s + (−0.963 − 0.266i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.753 − 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 − 0.990i)4-s + (0.309 − 0.951i)5-s + (0.936 + 0.351i)6-s + (0.393 − 0.919i)7-s + (−0.550 − 0.834i)8-s + (−0.550 + 0.834i)9-s + (−0.393 − 0.919i)10-s + (0.0448 + 0.998i)11-s + (0.936 − 0.351i)12-s + (0.0448 − 0.998i)13-s + (−0.309 − 0.951i)14-s + (0.983 − 0.178i)15-s + (−0.963 − 0.266i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.359 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.421726666 - 1.662992248i\)
\(L(\frac12)\) \(\approx\) \(2.421726666 - 1.662992248i\)
\(L(1)\) \(\approx\) \(1.793065307 - 0.7377540338i\)
\(L(1)\) \(\approx\) \(1.793065307 - 0.7377540338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.753 - 0.657i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.393 - 0.919i)T \)
11 \( 1 + (0.0448 + 0.998i)T \)
13 \( 1 + (0.0448 - 0.998i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.983 + 0.178i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (0.963 - 0.266i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (0.858 + 0.512i)T \)
47 \( 1 + (-0.473 + 0.880i)T \)
53 \( 1 + (-0.134 - 0.990i)T \)
59 \( 1 + (-0.936 + 0.351i)T \)
61 \( 1 + (0.393 + 0.919i)T \)
67 \( 1 + (-0.134 + 0.990i)T \)
73 \( 1 + (0.753 - 0.657i)T \)
79 \( 1 + (-0.550 - 0.834i)T \)
83 \( 1 + (0.936 - 0.351i)T \)
89 \( 1 + (0.134 + 0.990i)T \)
97 \( 1 + (-0.623 - 0.781i)T \)
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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

−31.48205095342456353738361956646, −30.69262036471953059200032421770, −29.930005886569432937015684928993, −28.76505962656303891005332233678, −26.68672517333787253107503315706, −26.01534466270648922298662951542, −24.837938126560218228418876502916, −24.22050341032349414728971284364, −22.98766858117915114935024136605, −21.81367133614022358184936705064, −20.95639325029349289574700273486, −19.04282952356042796996343551963, −18.349415160570875439408185900540, −17.05516410719266616358676986493, −15.464668346683408699810168150288, −14.32626083875039275593258843096, −13.826350002472791865689999003128, −12.313488628095399939902283122345, −11.34160315809575955826169872917, −9.00282379656888169053148473478, −7.84181119337116824360715573519, −6.582440509793559919785069750402, −5.667112175340865373093777542822, −3.43960833717551423293752840848, −2.24697994939779057431832814734, 1.31642630790054883284792900261, 3.22881635073818956959789150036, 4.58336335915316205052174676195, 5.371661006027495822254258425311, 7.73253579612720104876469346893, 9.53176032644421210142574479682, 10.19382076658291965634489381685, 11.654107004964403841789475288727, 13.08011318973442423360919069697, 14.013835967566637591846944880623, 15.15350534646059638454780470245, 16.338651431554210683873047193958, 17.70240803127799395718171725257, 19.670528604071052890824982936910, 20.57791109372072400952532226548, 20.8198566981256928933374810013, 22.31684026098311840257467970409, 23.268409902803763379271737704538, 24.61101556959895228486292069943, 25.62405135244756722005914482152, 27.359444699094723147984288194076, 27.86993644336076876461041732286, 29.13361584962574512478213856687, 30.2881364715862425267961822015, 31.360141264830232403867653439593

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