Properties

Degree 1
Conductor 137
Sign $0.153 + 0.988i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.602 + 0.798i)2-s + (0.995 + 0.0922i)3-s + (−0.273 + 0.961i)4-s + (0.798 + 0.602i)5-s + (0.526 + 0.850i)6-s + (−0.739 − 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.982 + 0.183i)9-s + i·10-s + (0.273 − 0.961i)11-s + (−0.361 + 0.932i)12-s + (−0.673 + 0.739i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 − 0.526i)16-s + (−0.932 − 0.361i)17-s + ⋯
L(s,χ)  = 1  + (0.602 + 0.798i)2-s + (0.995 + 0.0922i)3-s + (−0.273 + 0.961i)4-s + (0.798 + 0.602i)5-s + (0.526 + 0.850i)6-s + (−0.739 − 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.982 + 0.183i)9-s + i·10-s + (0.273 − 0.961i)11-s + (−0.361 + 0.932i)12-s + (−0.673 + 0.739i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 − 0.526i)16-s + (−0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.153 + 0.988i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.153 + 0.988i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.153 + 0.988i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (9, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.153 + 0.988i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.465801613 + 1.255378724i$
$L(\frac12,\chi)$  $\approx$  $1.465801613 + 1.255378724i$
$L(\chi,1)$  $\approx$  1.538357133 + 0.8791061977i
$L(1,\chi)$  $\approx$  1.538357133 + 0.8791061977i

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

−28.44714193366659087899720756061, −27.555921569267594422243355880629, −26.14036006518835808288390136041, −24.93060601327478089869099282068, −24.69496443438747522945033863942, −23.07014994228999084707090816631, −22.01696906517554002199841560973, −21.19978020260621074345355202829, −20.24175159315210443155345785590, −19.56784252964647155630561921826, −18.506519125544364736274407160170, −17.30897255779015775757984174025, −15.46886488330795546056872440765, −14.87445550553067707164979677394, −13.519807429702993287006323026149, −12.89145802971748992247452269273, −12.09553269848704448903154210617, −10.07785918136235871333984219329, −9.61536394579658828324704471973, −8.52428397699866379266921962903, −6.6526849531327049432885084242, −5.34488213165289165212395572673, −4.044947446524116658774574020141, −2.62807317935978071314431395207, −1.76933754409114129232026029786, 2.499316980376972957810150423230, 3.48825818840050473740700163581, 4.79583577806173431483820718819, 6.61600207424105198674766000621, 6.9883878335832363865743429580, 8.66038706114789666992290790882, 9.482434193550329914783720908038, 10.89872067228035941900014979773, 12.7490007945815701845737080699, 13.702724476643109834150235823989, 14.137599112473460021278667742305, 15.250082607535853229270160717345, 16.35802253389298418244648135494, 17.31197838995929009024630646781, 18.66081629547079508071681664116, 19.67460620734007590780582261203, 21.00175618440738621172471213001, 21.851052995152477380551436310260, 22.576250824791448850552706814360, 24.07239303560711950522001850765, 24.73152617702937850634852405773, 25.862988064948869848314295460183, 26.39268394526706207670151552477, 27.022725220766877660532608626407, 29.16217815991903762935728238294

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