Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.472 + 0.881i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 − 0.642i)2-s + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (0.396 − 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (−0.835 + 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.766 − 0.642i)2-s + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (0.396 − 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (−0.835 + 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.472 + 0.881i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.472 + 0.881i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.472 + 0.881i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3709, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.472 + 0.881i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2643433080 - 0.1581894142i$
$L(\frac12,\chi)$  $\approx$  $-0.2643433080 - 0.1581894142i$
$L(\chi,1)$  $\approx$  0.6638131724 - 0.6824246537i
$L(1,\chi)$  $\approx$  0.6638131724 - 0.6824246537i

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

−18.672926328400085592572762882785, −18.31262454432451753814826332137, −17.44196643584640017006626912196, −16.80047716441397656582937341955, −16.23657445833830769372003308459, −15.69357177225186841801010267410, −15.05825599105633982254392363073, −14.30615405165377409333573283486, −13.678002567856618714345611250862, −12.85696741958767003995395290672, −12.37510999508238924642055112836, −11.510357450585608123469369730601, −10.60829596151431024747390717358, −10.405210848376021053648159314498, −9.46582455858046547674974117890, −8.44707723022284159078748404100, −7.713855192930281275074043183226, −6.66333098695874159596158620690, −6.301851381501486466201034942575, −5.69229733669671143287517954264, −5.22910961206361553893521022636, −3.84055546653078326255969356027, −3.48462705445200527818648825844, −2.98424509163619676460223191950, −1.556985162322313994303635004, 0.07874423058597604358050983946, 0.982139511412462243420764215325, 2.00316065273316478022004160286, 2.38052254207589600398792915211, 3.71842913452319080277139603205, 4.47305092173746725466073429345, 5.07100740871252784842113218571, 5.84922127739885028763547805523, 6.39578188151568569618584501525, 7.03421131136038173140752453952, 8.071297487570756368066011386, 9.233086450929634104411892461555, 9.747105687758406762326661060125, 10.316395313546529964719011728319, 11.29272308299144960255482452653, 11.97282605316111843609527252456, 12.39720390713571703957914070757, 13.04152973063456886910334928549, 13.54017201318663387194325135275, 14.07345103154476262726898187081, 15.44271653862732105223247179552, 15.716073245988080910337639162429, 16.72038274411436052841636432683, 16.98979066854192926703717839285, 18.19109094168497926879937302572

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