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Properties

Label 1-71-71.37-r0-0-0
Degree $1$
Conductor $71$
Sign $0.951 - 0.308i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (0.623 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)12-s + (0.623 + 0.781i)13-s + 14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (0.623 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)12-s + (0.623 + 0.781i)13-s + 14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.951 - 0.308i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7005854572 - 0.1106987483i\)
\(L(\frac12)\) \(\approx\) \(0.7005854572 - 0.1106987483i\)
\(L(1)\) \(\approx\) \(0.7790124998 - 0.1509676048i\)
\(L(1)\) \(\approx\) \(0.7790124998 - 0.1509676048i\)

Euler product

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

−32.34045903916122924315773574966, −30.369606966479402034461946913752, −29.64421262535182178020173234280, −28.2814714875060010430252953800, −27.54027468112122415373115188843, −26.01509155640342860204230906406, −25.212152546680311686408939458, −24.16844768972328823079998767632, −22.930527238809775530075557560842, −22.50373928523690590594512713323, −20.78345922076141572697618652739, −19.05507198693743772788878749492, −17.92699059861400426790471883450, −17.11541734964673464248747387196, −16.47495536307196982050069774318, −14.760123770458890503920896441306, −13.53841349769065035291927671536, −12.6523416393900529742938166707, −10.566271860088312866421530530734, −9.781088979423175784187715373194, −7.912598327145820400555493075728, −6.64422204129508092241391219412, −5.84888523146463653708230688886, −4.390595547796402968533976708715, −1.2714832275738447444620019808, 1.63297725600770533116526031438, 3.49872685697830276290207900607, 5.21494795881166720366484835230, 6.2846602497051735150749592802, 8.85765667508235970274738317491, 9.63552694616610677515279426694, 10.91950269963046067785114074143, 11.88896796461440117476299856546, 13.038348736982285258359669177783, 14.40945683127487822988718834223, 16.2582978377192094825451366419, 17.25394749286459951516329364885, 18.294702889380371474941662945703, 19.21406307111344782389767245945, 21.15246418517548344383471267278, 21.50087977361961876849300299041, 22.37972076452108765203057940096, 23.6795729085259674539315203902, 25.32558696500511999610252619672, 26.450004431672962801853349989655, 27.84009144706576889531930942768, 28.33189534997118534828532749383, 29.42011010284004011598476394369, 30.06028173416837434280598827232, 31.75202199788214580261472032865

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