Properties

Label 1-71-71.36-r0-0-0
Degree $1$
Conductor $71$
Sign $-0.443 - 0.896i$
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  + (−0.995 + 0.0896i)2-s + (0.134 − 0.990i)3-s + (0.983 − 0.178i)4-s + (−0.809 − 0.587i)5-s + (−0.0448 + 0.998i)6-s + (0.858 − 0.512i)7-s + (−0.963 + 0.266i)8-s + (−0.963 − 0.266i)9-s + (0.858 + 0.512i)10-s + (−0.550 + 0.834i)11-s + (−0.0448 − 0.998i)12-s + (−0.550 − 0.834i)13-s + (−0.809 + 0.587i)14-s + (−0.691 + 0.722i)15-s + (0.936 − 0.351i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.134 − 0.990i)3-s + (0.983 − 0.178i)4-s + (−0.809 − 0.587i)5-s + (−0.0448 + 0.998i)6-s + (0.858 − 0.512i)7-s + (−0.963 + 0.266i)8-s + (−0.963 − 0.266i)9-s + (0.858 + 0.512i)10-s + (−0.550 + 0.834i)11-s + (−0.0448 − 0.998i)12-s + (−0.550 − 0.834i)13-s + (−0.809 + 0.587i)14-s + (−0.691 + 0.722i)15-s + (0.936 − 0.351i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.443 - 0.896i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ -0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2743564604 - 0.4417323020i\)
\(L(\frac12)\) \(\approx\) \(0.2743564604 - 0.4417323020i\)
\(L(1)\) \(\approx\) \(0.5340511467 - 0.3188941252i\)
\(L(1)\) \(\approx\) \(0.5340511467 - 0.3188941252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (0.134 - 0.990i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.858 - 0.512i)T \)
11 \( 1 + (-0.550 + 0.834i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.691 - 0.722i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.473 + 0.880i)T \)
31 \( 1 + (0.936 + 0.351i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (0.753 - 0.657i)T \)
47 \( 1 + (0.134 + 0.990i)T \)
53 \( 1 + (0.983 + 0.178i)T \)
59 \( 1 + (-0.0448 - 0.998i)T \)
61 \( 1 + (0.858 + 0.512i)T \)
67 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (-0.995 + 0.0896i)T \)
79 \( 1 + (-0.963 + 0.266i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (0.983 + 0.178i)T \)
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

−31.90161033884429983083450764952, −31.03041535143049822727432131650, −29.75302841475549461345453025738, −28.34683238240421562695905104378, −27.57886536142750027213434309846, −26.71395597352788158524659843687, −26.07458655254404232273373445085, −24.59986593123917916822286899583, −23.37782095390246945782989867250, −21.56698868315337446205471507883, −21.17053747917586156497238351943, −19.5792216067779698592638584935, −18.8846308091737290492561763598, −17.45686692492822756425928743068, −16.2856634477227373712733668389, −15.305937787469634083592992620963, −14.453659350394601508833293599382, −11.89086151193674699150793186598, −11.10719635699244295756961296747, −10.08044276575102582417967599378, −8.58875795609135988819532900116, −7.82993297408954021332203843912, −5.91318104635530827601743654321, −3.99056666197731922321307032383, −2.469333142974235762832667278905, 0.83246353773661799190576449488, 2.54395100325912336790424060674, 4.99910335482711658314285265839, 7.03669402055883212159742083307, 7.79249673321695677402922241259, 8.70848318228836756365073634418, 10.4940325707004784674135873513, 11.785967092443352260317181350, 12.69799952229630780154241644153, 14.48677824660621508647368429239, 15.680562021168552093429592577239, 17.152630012622181044566588636873, 17.86161386232138369954524536518, 19.03877514164868266931657276899, 20.17417666589075809753756311653, 20.620643402561436493029768090932, 23.06440443617418764869417143036, 24.00788255428581717340382223479, 24.74842780285572088813761286295, 25.88195208167996760682254748664, 27.16463634054971564803796019800, 27.978473563009613787239652579917, 29.097480198590414272343436062591, 30.2030026171865830280785679070, 31.008403896375952399163872349059

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