Properties

Degree 1
Conductor $ 3 \cdot 23 $
Sign $0.938 - 0.343i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.959 + 0.281i)8-s + (0.142 − 0.989i)10-s + (0.415 − 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.841 − 0.540i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.959 + 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯
L(s,χ)  = 1  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.959 + 0.281i)8-s + (0.142 − 0.989i)10-s + (0.415 − 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.841 − 0.540i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.959 + 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.938 - 0.343i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.938 - 0.343i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(69\)    =    \(3 \cdot 23\)
\( \varepsilon \)  =  $0.938 - 0.343i$
motivic weight  =  \(0\)
character  :  $\chi_{69} (44, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 69,\ (0:\ ),\ 0.938 - 0.343i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8137070293 - 0.1443500053i$
$L(\frac12,\chi)$  $\approx$  $0.8137070293 - 0.1443500053i$
$L(\chi,1)$  $\approx$  0.8899987882 - 0.1783700740i
$L(1,\chi)$  $\approx$  0.8899987882 - 0.1783700740i

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

−32.5440326308523802384565577873, −30.94384749640697463037889074420, −29.57837374853558648901023464798, −28.45634066315016718725780031915, −27.43292586640502810916094668704, −26.32920449881814882683333542651, −25.27555787193350820013161722797, −24.499683093663891267180960927984, −23.28765842074798383139453645470, −22.26466565918762262961032798013, −20.57441416690903346094363877173, −19.67336729869393090830112664476, −17.85472301328270956332292553345, −17.38923505056402476351618676841, −16.29536244000798956853389132624, −14.88484396162371630586203736460, −13.8055479701953317971633802768, −12.73031864122734976037167146390, −10.48419708069232508111908221468, −9.65231083170375161879861039626, −8.241228521904833116773598825808, −6.97993329426242430598678003245, −5.62556278685487834258301686463, −4.31714857277121455719485055966, −1.44832072438346395236575550296, 1.91991516442511312934350038573, 3.14446597763067472307726583575, 5.12900997259858663300045638222, 6.8116732698900913621768697865, 8.77079079222485166252144346309, 9.48303174191687701011816647827, 11.00896056530073663784984794030, 11.86866868758031517432604026402, 13.411202457013816508026975816819, 14.344664753290032013524919808325, 16.179385702720742440464692085956, 17.56568009291094032797394806449, 18.44884710597772030997069226663, 19.31668730813322527394509184423, 20.80559740804322202611998315545, 21.893515339033460014200975964208, 22.237290929986178739752452267194, 24.2392821159529998611334747988, 25.48590216067009675586482521466, 26.49108486911712530258013159322, 27.515661492908150027673867364197, 28.86474140896446252732141626495, 29.30966221402373428902859383751, 30.64207638158385601014502477587, 31.4386147791410499595128836205

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