↑

Properties

Label 1-629-629.169-r0-0-0
Degree $1$
Conductor $629$
Sign $-0.760 - 0.649i$
Analytic cond. $2.92106$
Root an. cond. $2.92106$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + 6-s + (0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (0.939 − 0.342i)15-s + (−0.939 + 0.342i)16-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + 6-s + (0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (0.939 − 0.342i)15-s + (−0.939 + 0.342i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $-0.760 - 0.649i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ -0.760 - 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1200382719 - 0.3256016830i\)
\(L(\frac12)\) \(\approx\) \(0.1200382719 - 0.3256016830i\)
\(L(1)\) \(\approx\) \(0.4685234198 - 0.1360254502i\)
\(L(1)\) \(\approx\) \(0.4685234198 - 0.1360254502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.50652078025107453095450385212, −22.89912477718758586615345845079, −21.8415993344655738159213593068, −20.54867639605228197195461158982, −19.51060196409794890866251682503, −19.14933203749249254099082613299, −18.130809373011039987177119627107, −17.48552007171763875563223166317, −16.89088212376625843773881054020, −15.94526235265620217379708699177, −15.03551401532214407746323727062, −14.35004164493345818985755132375, −13.305570856219567841213313185177, −11.75985159991929100555575607284, −11.589840805840514748118151112516, −10.64559971137022256642556661418, −9.62520298844799793903890248080, −8.33275962440553963120484155497, −7.67373947867960576901076420885, −6.94508871905502346578052538036, −6.29499016605877537774636155593, −4.86125212341753783044196833167, −4.32006030158086171838868044106, −2.16814050230947942586076203372, −1.27880209035915288909857079205, 0.28326758026343881823030775641, 1.46389076174222849658971114528, 3.09449750361237476742102718081, 3.99821665253212395271510502656, 4.83898229634365972459158334024, 6.00915409569586199232350720286, 7.29312779034001044693641578481, 8.47579845103903972523316057238, 8.66226486163193668735422292509, 10.13566045071852861624400142114, 10.76773568803291860926861521739, 11.59005367897484204209242325894, 12.07154020990921462529583724215, 12.92154132821965419169835777599, 14.53853502826467555495685555575, 15.37500669397541251853079169701, 16.26260717030653552445786604351, 16.86987757123793752932596651719, 17.66714407566380076836141139226, 18.50728428146697993837036194198, 19.29251269351130937148408866497, 20.3900433943609425826476471728, 20.77842731658600640672685227675, 21.78525863034671361723453410146, 22.36832765902788147137600233379

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