Properties

Degree 1
Conductor $ 17 \cdot 59 $
Sign $-0.615 - 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + i·3-s + 4-s + i·5-s + i·6-s i·7-s + 8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s i·14-s − 15-s + 16-s + ⋯
L(s,χ)  = 1  + 2-s + i·3-s + 4-s + i·5-s + i·6-s i·7-s + 8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s i·14-s − 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.615 - 0.788i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.615 - 0.788i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1003\)    =    \(17 \cdot 59\)
\( \varepsilon \)  =  $-0.615 - 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{1003} (353, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1003,\ (1:\ ),\ -0.615 - 0.788i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4148418178 + 0.8502104973i$
$L(\frac12,\chi)$  $\approx$  $-0.4148418178 + 0.8502104973i$
$L(\chi,1)$  $\approx$  1.323786964 + 0.7283457939i
$L(1,\chi)$  $\approx$  1.323786964 + 0.7283457939i

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

−21.100064125318241129398844814114, −20.21796424666414659885768458566, −19.320918147565562759833499128506, −19.006862998829016975963832505763, −17.7177320163264863366990764995, −16.86564992958390201477987922755, −16.2736478381286470590028288695, −15.238458841902144875455380095617, −14.47662080054493771191985357329, −13.66053880040707347571708869336, −12.83454934656239670465050445194, −12.38764256572699347984789427001, −11.76407906381109865986286625438, −10.919429165969664364682779469534, −9.52288493759619394456137427823, −8.37046541844464357143157449691, −8.042260724068743373202904554231, −6.65891692557810196413265831251, −6.06723264462381166849174479787, −5.24593082676846650763587591784, −4.48607941044643019195395779421, −3.05229172564148120244953895910, −2.333778529753082873579383041502, −1.385674804122430591248537959318, −0.117296787812000493518707774413, 1.92190482240001963205679897700, 2.819062091713868229757268380311, 3.79161954125462858020879283721, 4.34026401817333486943098576660, 5.23741217738202059277769782332, 6.26425399386816929514427418868, 7.20632793422403169070936476557, 7.707657565201551448645050786325, 9.394200450380607061222463248842, 10.21957556663421338064645624729, 10.70748991458535643757906650744, 11.458219429607548220292223999436, 12.415626592418086119019095063339, 13.40738416605192521464869334112, 14.3783105950326992176061512918, 14.702194776406429047168698802735, 15.39360642773816461124436622717, 16.25215293114700932009382681074, 17.18289642775097859968529319540, 17.66290567372798590127237164700, 19.355791077674398325120804389966, 19.715694351251793455288307662633, 20.657369134349167541221871558, 21.24158060729709250497356388417, 22.18659843071947377443981458778

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