Properties

Label 1-71-71.10-r0-0-0
Degree $1$
Conductor $71$
Sign $-0.563 - 0.826i$
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  + (0.858 − 0.512i)2-s + (−0.691 − 0.722i)3-s + (0.473 − 0.880i)4-s + (−0.809 − 0.587i)5-s + (−0.963 − 0.266i)6-s + (−0.995 + 0.0896i)7-s + (−0.0448 − 0.998i)8-s + (−0.0448 + 0.998i)9-s + (−0.995 − 0.0896i)10-s + (0.936 + 0.351i)11-s + (−0.963 + 0.266i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.134 + 0.990i)15-s + (−0.550 − 0.834i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (−0.691 − 0.722i)3-s + (0.473 − 0.880i)4-s + (−0.809 − 0.587i)5-s + (−0.963 − 0.266i)6-s + (−0.995 + 0.0896i)7-s + (−0.0448 − 0.998i)8-s + (−0.0448 + 0.998i)9-s + (−0.995 − 0.0896i)10-s + (0.936 + 0.351i)11-s + (−0.963 + 0.266i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.134 + 0.990i)15-s + (−0.550 − 0.834i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ -0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4732078119 - 0.8950102589i\)
\(L(\frac12)\) \(\approx\) \(0.4732078119 - 0.8950102589i\)
\(L(1)\) \(\approx\) \(0.8577348802 - 0.7205995562i\)
\(L(1)\) \(\approx\) \(0.8577348802 - 0.7205995562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.858 - 0.512i)T \)
3 \( 1 + (-0.691 - 0.722i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.995 + 0.0896i)T \)
11 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.134 - 0.990i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.983 + 0.178i)T \)
31 \( 1 + (-0.550 + 0.834i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.393 + 0.919i)T \)
47 \( 1 + (-0.691 + 0.722i)T \)
53 \( 1 + (0.473 + 0.880i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (-0.995 - 0.0896i)T \)
67 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.858 - 0.512i)T \)
79 \( 1 + (-0.0448 - 0.998i)T \)
83 \( 1 + (-0.963 + 0.266i)T \)
89 \( 1 + (0.473 + 0.880i)T \)
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

−32.39828485040108221611557137433, −31.15196279100106452233606213171, −30.06626314811157784422628535629, −28.975186830181670646158736788918, −27.59332852929106784938673218091, −26.491434358693756607632256673925, −25.68174351676052905849210894738, −24.0408666563966570425680970225, −23.00251469888533277112526010703, −22.504267497528462547354398511006, −21.4782286909512139676369317095, −20.12913786719402155772185090499, −18.68008033691442576691631263708, −16.90289173744616082955921770640, −16.21577340015281566659971038336, −15.25010360536927764937588813809, −14.159664481822248844405997729734, −12.4809886017335507961152741556, −11.56964100476287674895262994976, −10.36109399982570684507443036130, −8.5138309412885460377494696581, −6.68911052415497478836036524761, −6.01775519316817291415837650346, −4.06579946118368735705459201803, −3.485356020490125895473834653133, 1.11391127930369398696664897707, 3.24974655157507772381136511187, 4.76823621762235705251646180147, 6.14614222746211719757850112390, 7.27370531334757447162049620228, 9.30402566431042365181749123521, 11.03238994073998102683527729593, 11.98818084750195147579091271617, 12.78789548071733781029131528327, 13.81896843702951481942506119549, 15.634805321421686655063365016483, 16.39247767224229743619463923465, 18.11583538627741893841930874565, 19.51564474088557698558017762567, 19.91517296082412635926050543967, 21.62914271381529403184153099933, 22.9454892412738457443376113571, 23.18550963814036982529096614348, 24.51298411398796231053162846998, 25.37519231455654217604963836775, 27.61679402363694449772124870485, 28.276230379843738043666286152711, 29.2892456426333973147924525970, 30.26572411357450062331934712308, 31.17325489188524545912344591687

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