Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.273 - 0.962i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 + 0.342i)2-s + (−0.396 + 0.918i)3-s + (0.766 − 0.642i)4-s + (−0.230 + 0.973i)5-s + (0.0581 − 0.998i)6-s + (−0.286 − 0.957i)7-s + (−0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (0.286 + 0.957i)12-s + (−0.286 − 0.957i)13-s + (0.597 + 0.802i)14-s + (−0.802 − 0.597i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 + 0.342i)2-s + (−0.396 + 0.918i)3-s + (0.766 − 0.642i)4-s + (−0.230 + 0.973i)5-s + (0.0581 − 0.998i)6-s + (−0.286 − 0.957i)7-s + (−0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (0.286 + 0.957i)12-s + (−0.286 − 0.957i)13-s + (0.597 + 0.802i)14-s + (−0.802 − 0.597i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.273 - 0.962i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.273 - 0.962i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.273 - 0.962i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (430, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.273 - 0.962i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3656796075 - 0.2763355984i$
$L(\frac12,\chi)$  $\approx$  $0.3656796075 - 0.2763355984i$
$L(\chi,1)$  $\approx$  0.5227291373 + 0.1167377346i
$L(1,\chi)$  $\approx$  0.5227291373 + 0.1167377346i

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

−18.50105943814779104816957470892, −18.100214592301023313878967906068, −17.28546960797868671644606955657, −16.76416566064319535665271051843, −16.13349237836789851073618287646, −15.54373734776421250786447409659, −14.55416332216390225171262481054, −13.59655946804254273217183606204, −12.610791112163499867701224871846, −12.454985682454130597349681586742, −11.71077432196680675486419828299, −11.424584325009003907693413474433, −10.146736940327523342372967085384, −9.47603812659008093160433517059, −8.91847163479675550237637670076, −8.1825421308664157699979816076, −7.60378719240210452493473715521, −6.68549548084056497779142383871, −6.30483749203354690612143329017, −5.08673234611696406484103228477, −4.588560753128356241393872700282, −3.19541921482602087945877951839, −2.38604859650340638023189707917, −1.69177904196854004969890057760, −1.00271433306684376599468152435, 0.24761912636876306864377387894, 1.01090625430294278939440680756, 2.5611832704508540909235900171, 3.23111282783144490804860216022, 3.86017968535000115305681587486, 5.032494742708156388883405219675, 5.72311707807862274375244710791, 6.69444505957973388431807194157, 6.8511974787231561290421642735, 8.039272105741641884936012527981, 8.50161671024968497826571333734, 9.50597619538519099407150225522, 10.2080547646621492858314254311, 10.527153154911850962181631596672, 11.23340575783578218741133270009, 11.57349526621817497891554880875, 12.861492639757211173559938606035, 13.841506718740575545549759066501, 14.48841405281466633775746062430, 15.28491277064353521550828405449, 15.568276122359974690276339470829, 16.34931070537929251092863767393, 17.1452715753682671093915250473, 17.40996714879986154658257871569, 18.13472344908971595018787136340

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