Properties

Degree 1
Conductor 71
Sign $0.951 + 0.308i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.951 + 0.308i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (48, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.951 + 0.308i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7005854572 + 0.1106987483i$
$L(\frac12,\chi)$  $\approx$  $0.7005854572 + 0.1106987483i$
$L(\chi,1)$  $\approx$  0.7790124998 + 0.1509676048i
$L(1,\chi)$  $\approx$  0.7790124998 + 0.1509676048i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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