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Properties

Label 1-71-71.5-r0-0-0
Degree $1$
Conductor $71$
Sign $-0.996 - 0.0826i$
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  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.996 - 0.0826i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ -0.996 - 0.0826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01217865317 + 0.2940822906i\)
\(L(\frac12)\) \(\approx\) \(0.01217865317 + 0.2940822906i\)
\(L(1)\) \(\approx\) \(0.3279939491 + 0.3013909575i\)
\(L(1)\) \(\approx\) \(0.3279939491 + 0.3013909575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + T \)
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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

−30.84316422784387602605525471284, −29.44188413440415434657523332918, −29.05588734887222928396899183080, −28.270626298435431148472996323949, −26.97654618512460453127996924534, −25.83793588099950828161394556042, −24.56878731229704368417678075988, −23.61652841592640258463755258470, −22.107052927082502602475076041732, −21.14083876243748203358200505917, −19.70697062358885195459312496618, −19.00678490688252300655201549452, −17.53146842179754982766465049974, −16.87539312833359050016607531921, −15.994807286412276136424074482821, −13.17370812945212727383714824447, −12.90165672434144851771101690380, −11.42748969414471944414479328525, −10.31788352627404538595631311296, −9.02465680567623997267246494324, −7.6010874255193518571521793890, −6.29594447348181379388042542827, −4.48032212480124763818385658072, −2.250321221328391446976105885199, −0.461585652060805022379748845854, 2.6544439406429678036099047441, 5.089587318776838477140710822205, 6.250281633197230745848248366028, 7.26206774242644784553441098714, 9.24140372569924570449288971359, 10.17550958885640659928465105446, 11.022876887653115985261621579047, 12.703067024566733540706402934404, 14.76801219014582019705235327553, 15.44875233481685793806650860563, 16.55610665870184007638924725086, 17.81752057232082059589474127901, 18.42871562099260696942333032832, 19.801334514721464651185491578951, 21.41234166715115041080514504548, 22.619125342856281478072607938229, 23.27715635014118741602691203487, 24.932724889993186109226592174917, 25.89832134675377172596977063007, 26.82941011191079122371881063104, 27.74670056892099314938584306531, 29.07228449423199476328158650325, 29.35177152209644115739172154649, 31.42661117480413602316715816323, 32.69374853076303574987829519823

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