Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.0592 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 − 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (0.396 − 0.918i)5-s + (−0.957 + 0.286i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (−0.549 + 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.993 − 0.116i)12-s + (−0.116 − 0.993i)13-s + (0.998 + 0.0581i)14-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s,χ)  = 1  + (−0.984 − 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (0.396 − 0.918i)5-s + (−0.957 + 0.286i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (−0.549 + 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.993 − 0.116i)12-s + (−0.116 − 0.993i)13-s + (0.998 + 0.0581i)14-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0592 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0592 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.0592 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (380, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.0592 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.6234133836 - 0.6615068307i$
$L(\frac12,\chi)$  $\approx$  $-0.6234133836 - 0.6615068307i$
$L(\chi,1)$  $\approx$  0.6326277369 - 0.5062864135i
$L(1,\chi)$  $\approx$  0.6326277369 - 0.5062864135i

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

−18.77294795726662461470249864068, −18.42313096338371578890748318947, −17.34400773213360165532654890971, −16.809457979416806138436934708092, −16.12367670364081430723823404393, −15.282008835844590302572856395868, −14.91429042359885642650616877754, −14.40557537650379045031177355092, −13.21519965997317621012129420685, −12.96403558885943407254447450568, −11.65141882209499364213654992502, −10.90855084563474638739557768440, −10.25136278857787667938401938280, −9.70588467867807488225698320860, −9.31800580456896964973332579858, −8.513987123894723832934532458664, −7.47661103410611076402196466607, −7.18400878245732537128362268767, −6.41405630323575092212408880157, −5.62200497607723477197837740188, −4.44039918409239511637667990143, −3.52541439148171782361350808003, −2.72998344840465650749544549520, −2.24805775464391022450276581533, −1.39795088941011571877272784194, 0.201577015250631049117110786142, 0.688960108455164824381134743579, 1.63551834325856991690066498066, 2.463546395046419092287451900047, 3.23356507227494373867123747222, 3.68891834040454609766872485926, 5.31296672220377374058911910029, 5.81969977202808066439612581909, 6.84558208835939267078247125029, 7.41421416342724650337446822629, 8.33089961845645964677810265507, 8.693531129380359781684869855101, 9.30530444363456796433533417701, 10.099490408072270045115793390382, 10.48113801619528792340830056054, 11.72951512785817354932622675325, 12.47240855258682285114316078333, 12.95227942564480088557206275828, 13.396339601804634065126126045660, 14.44896015944163041416784691721, 15.24120005059193533533054567081, 15.96299538158644710397357937668, 16.49793307030151178554720424562, 17.06079597817273806018839829523, 18.04643186676400909069338803477

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