Properties

Degree 1
Conductor 71
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{71} (70, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.103519213$
$L(\frac12,\chi)$  $\approx$  $4.103519213$
$L(\chi,1)$  $\approx$  2.609869177
$L(1,\chi)$  $\approx$  2.609869177

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

−31.580496763404839077015148075897, −30.5667367046612922159597786796, −29.30063365876838953238927122884, −28.85034318546807265152895296853, −26.55428610706120910245589081450, −25.781022323353333227390685058355, −24.881327341465210261241708637335, −23.94052003187434134468456387017, −22.29323285862342181606200346449, −21.64819249454551935669004501804, −20.42359925551303459734122628552, −19.644436270149691799299472550432, −18.14418164274828099915935797125, −16.36111705724857734309410567016, −15.3876649212004167158047334910, −14.11267363061412232559542704023, −13.331983299432117302324992030021, −12.48005273655691524168692086390, −10.39998094941083720032644625035, −9.4619945689777965060402749771, −7.589853268223378404160937901117, −6.330337949316219785645225721908, −4.83691095115137534925409333274, −3.10856692743901130476400230901, −2.17958092383814282217169409664, 2.17958092383814282217169409664, 3.10856692743901130476400230901, 4.83691095115137534925409333274, 6.330337949316219785645225721908, 7.589853268223378404160937901117, 9.4619945689777965060402749771, 10.39998094941083720032644625035, 12.48005273655691524168692086390, 13.331983299432117302324992030021, 14.11267363061412232559542704023, 15.3876649212004167158047334910, 16.36111705724857734309410567016, 18.14418164274828099915935797125, 19.644436270149691799299472550432, 20.42359925551303459734122628552, 21.64819249454551935669004501804, 22.29323285862342181606200346449, 23.94052003187434134468456387017, 24.881327341465210261241708637335, 25.781022323353333227390685058355, 26.55428610706120910245589081450, 28.85034318546807265152895296853, 29.30063365876838953238927122884, 30.5667367046612922159597786796, 31.580496763404839077015148075897

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