Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.954 - 0.298i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.642 + 0.766i)2-s + (0.973 + 0.230i)3-s + (−0.173 − 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.802 + 0.597i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (0.957 − 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (0.998 + 0.0581i)13-s + (−0.727 − 0.686i)14-s + (−0.686 − 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s,χ)  = 1  + (−0.642 + 0.766i)2-s + (0.973 + 0.230i)3-s + (−0.173 − 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.802 + 0.597i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (0.957 − 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (0.998 + 0.0581i)13-s + (−0.727 − 0.686i)14-s + (−0.686 − 0.727i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.954 - 0.298i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.954 - 0.298i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.954 - 0.298i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2711, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.954 - 0.298i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2093472660 + 1.369281904i$
$L(\frac12,\chi)$  $\approx$  $-0.2093472660 + 1.369281904i$
$L(\chi,1)$  $\approx$  0.8403983319 + 0.5100718671i
$L(1,\chi)$  $\approx$  0.8403983319 + 0.5100718671i

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

−18.08934428470812433038029475177, −17.46837831121401086150966951647, −16.59941132099930868186449488979, −15.97292498816775478474353397937, −15.18930158751868240127105289331, −14.35426303581187076760149655856, −13.834277962936978104578524129464, −12.99345729276050829539885586780, −12.56411946428790965404208396312, −11.526622887507967981961896291934, −10.787180609540170201062856967036, −10.66300131931385026512023528537, −9.38740310265752990662675878400, −8.96779445036844839135424513468, −8.24076640326246850067263064195, −7.5858816677685628958007233380, −6.94442102682699726689452955986, −6.415854073500048075835486919800, −4.59456805940528868132642796415, −3.8513314774411485036758662037, −3.58194660010156090062688053573, −2.838654875959888823734443117467, −1.760247513377789792393433839269, −1.06005446247662373651570059941, −0.240396756119160475556475218221, 1.15358239299608923456758119366, 1.666558215667702438622064499004, 2.81010242171725265632651293820, 3.82240507787336193425317059672, 4.35311600404774478673205032501, 5.37110128551832768208366690183, 5.9628620240526470584537638507, 7.20711819617552021882086715474, 7.4396358000375173809593633805, 8.47433089272106558693538262683, 8.84810150463295598116415461094, 9.31858358959941258321583730669, 9.97133741055196460726648126011, 11.205400113408826258277527640789, 11.60135577738888413258075152037, 12.644013781345333862411406479962, 13.37995660836848703436817118948, 14.122184677060403430869127373225, 14.96071633676470765061343573807, 15.21253814464995640484089566932, 16.0082238037332297882413643231, 16.35378023929694761175268449529, 17.15566398370446938254428735397, 18.28254509911583045231930369611, 18.63094995289960438791094142002

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