Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.850 + 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s + 11-s i·12-s + i·13-s + 14-s + 16-s + i·17-s + i·18-s − 21-s i·22-s + ⋯
L(s,χ)  = 1  i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s + 11-s i·12-s + i·13-s + 14-s + 16-s + i·17-s + i·18-s − 21-s i·22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.850 + 0.525i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.850 + 0.525i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.850 + 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (37, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (0:\ ),\ 0.850 + 0.525i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8335455634 + 0.2367928266i$
$L(\frac12,\chi)$  $\approx$  $0.8335455634 + 0.2367928266i$
$L(\chi,1)$  $\approx$  0.9275494902 + 0.03930266770i
$L(1,\chi)$  $\approx$  0.9275494902 + 0.03930266770i

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

−30.159539533502942106939805301950, −29.32007424856168673334435438360, −27.72080870424319393218529402716, −26.90740799598479298280669375659, −25.53824299489389763113880900488, −24.9840559620957045834193153098, −23.86792426730969073671914285983, −23.089314416447147025745560782001, −22.21305171946824332234353308739, −20.26619714930276074581275285394, −19.33785107585546664480168691269, −18.012589172162846059720223963187, −17.33805618815856277020779392216, −16.314833696081505911274624189358, −14.78587393485959195097248721811, −13.81947834476811814622826063457, −13.042813107166041780008145601766, −11.62742805489271548170352145934, −9.89228135302715512590499703717, −8.46165874755401990785933012119, −7.373997355615696927886329510245, −6.59079200056891894465715183997, −5.197480674927745228047117036189, −3.49212982088035943518449251321, −1.00956596267739874773536933108, 2.09130176791743641415888366341, 3.598257768466807782480451106363, 4.669117732689184098467673418022, 6.07129657668141576993945810261, 8.63040877833238821619430856974, 9.19965023350534065310197560639, 10.446351358402905095988073523, 11.56102369892736197003156875931, 12.419085983319133485610991208081, 14.14325425392052788927733653922, 14.91716989304318759081223356987, 16.414074283471157200423545439657, 17.506881702470753098728565392443, 18.87552333710753681789562671057, 19.77064914948801266731111640320, 20.95879848065143768132890063917, 21.82134482732437452447833715440, 22.34067943959872757161917810414, 23.72814105388736099283558868762, 25.33401580375142739296972277056, 26.48626182142575535012291234574, 27.39556456309818443045846841691, 28.31428975809978745350478652463, 28.88748078081119860243758871447, 30.44794766237577854535476925315

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