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Properties

Label 1-71-71.20-r0-0-0
Degree $1$
Conductor $71$
Sign $0.791 - 0.611i$
Analytic cond. $0.329722$
Root an. cond. $0.329722$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + 5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)12-s + (−0.222 + 0.974i)13-s + 14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + 5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)12-s + (−0.222 + 0.974i)13-s + 14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7647409886 - 0.2610686514i\)
\(L(\frac12)\) \(\approx\) \(0.7647409886 - 0.2610686514i\)
\(L(1)\) \(\approx\) \(0.8688004713 - 0.1528414261i\)
\(L(1)\) \(\approx\) \(0.8688004713 - 0.1528414261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−31.88704636703430292204308515657, −30.54828537554864316661083777095, −29.47804516097162603489613904247, −28.34883460611041806722399964763, −27.648692439161607685810335620287, −26.12199046346034153124621863380, −25.68908839474883167252842502861, −24.88261903671017253203665626516, −22.45625165354025592952252350569, −21.6879886838722487930890901835, −20.55040193096449381868972883197, −19.79606111966092398759376036552, −18.44453194843035702165437113922, −17.34373504500783982675833756172, −16.13223566698412725557367127012, −15.12704531217669936100472144974, −13.46062960184485283118839185090, −12.269267779776899835130369267515, −10.2982508794360951743055941424, −9.86243944652772435179431239361, −8.834048825079180803660652624486, −7.31513403442771159619706470908, −5.46805659152052212659935709434, −3.32111464842679515047381373338, −2.22359683343568948642373168060, 1.39769212206860872549173020730, 2.9767362587563237185222722479, 5.898844985643792274078409156529, 6.750444475949214418432659020836, 8.10965216686621373139984254221, 9.349974428816980549980338035322, 10.230783245381269056256052873563, 12.10508974036465045027408784989, 13.73174615731396510125653113257, 14.2945540925180414895460904521, 16.14869198778600455137697272680, 17.03548450684688832886313584701, 18.395665553292255069092834830869, 19.02781612228181607840990940365, 20.17054703618761644492410595914, 21.36598665849964875465413689674, 23.216379451209307915185694855740, 24.334607735641673530459300994826, 25.197125743383915000930742182564, 26.10770094088894451321481275244, 26.75105831998336414958816723612, 28.68169601564951958134435858114, 29.26583500370182533113843643274, 30.072424322270348347449452022634, 31.89290470982637768060013817161

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