Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.414 + 0.909i$
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 + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (−0.835 + 0.549i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.973 + 0.230i)13-s + (0.993 + 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.939 + 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (−0.835 + 0.549i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.973 + 0.230i)13-s + (0.993 + 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.414 + 0.909i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.414 + 0.909i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.414 + 0.909i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (156, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 0.414 + 0.909i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.710789090 + 1.742996897i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.710789090 + 1.742996897i\)
\(L(\chi,1)\)  \(\approx\)  \(1.706349713 + 0.6611458318i\)
\(L(1,\chi)\)  \(\approx\)  \(1.706349713 + 0.6611458318i\)

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

−18.37524619512204066626584757148, −17.73423260821901554538901513761, −17.23134397745275748354596806748, −16.18532493002802981692124555944, −15.519423550811370695761223878699, −14.567090498887213298220352085905, −14.17763741126870168061465199292, −13.66117112558807156888480754047, −12.77247844804276959208682926203, −12.19975735595057400063912094151, −11.675459460984220694824728996617, −10.91873942477478369199501479017, −10.19739344681802645242914682128, −9.86808218653666948681988115421, −8.29916424707245423392301943460, −7.55774553469461731977498262314, −7.061068212372297231808108140869, −6.163539183039504182425957672878, −5.55123092490706991802441105279, −5.04002155603423685122122773069, −4.31137926226363052505981608764, −2.89067040313263985362795120955, −2.33027503270494151657857015426, −1.91942177063937784932606547402, −0.79589411299084394994677062036, 0.92283062669370081703538296240, 2.09493750397919123521700190523, 2.76141680892769488603989187768, 3.98283655135204787616674471830, 4.52873917639725856913603384195, 5.16628390989848171762965381854, 5.56270309701783465826962157818, 6.36168153867717023371129818560, 7.21376620388918500148004728101, 8.2771084529249349652682976882, 8.6025499350324771168996580170, 9.91361923879594409159145500610, 10.30717481316005650021313407171, 11.19479126238311652681001654049, 11.740045582218988873595279610274, 12.51177291367897530206249690269, 13.11821286613720368484418489614, 13.96380854309014250598495992750, 14.469614547698682688834176711590, 15.23059908188920963603296891578, 15.83558822961690198035928357026, 16.50545248952314941932870339931, 17.207166057950315612940899844401, 17.49395055527697425510485321712, 18.1918042840614136340055841721

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