Properties

Degree 1
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.713 - 0.700i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s − 11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s + (−0.5 − 0.866i)26-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.866 + 0.5i)32-s + (−0.5 − 0.866i)34-s + ⋯
L(s,χ)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s − 11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s + (−0.5 − 0.866i)26-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.866 + 0.5i)32-s + (−0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.713 - 0.700i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.713 - 0.700i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.713 - 0.700i$
motivic weight  =  \(0\)
character  :  $\chi_{315} (47, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 315,\ (1:\ ),\ -0.713 - 0.700i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.08071060372 + 0.1975032300i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.08071060372 + 0.1975032300i\)
\(L(\chi,1)\)  \(\approx\)  \(1.121562116 + 0.4229360452i\)
\(L(1,\chi)\)  \(\approx\)  \(1.121562116 + 0.4229360452i\)

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

−24.114583969548096395300022123464, −23.59610718748116673535926760488, −22.46581105245765240656279067042, −21.84073369699866423435298571144, −20.85653048779239633740558203070, −20.197575237561426932097500043, −19.10012129201802095299595885482, −18.449207205977855727997881080757, −17.056111554533869435561044801629, −16.04687588813453781056014390988, −15.04440250764646583313177100782, −14.36476323535500482431029158904, −13.193663775309486658368739275864, −12.59540836129537419563435535982, −11.53623535759788986105778051338, −10.56922893430922413450692186719, −9.802108449364017953139302600862, −8.40206945396823526913539171324, −7.093215130096852053402998804811, −6.08593448064226524540438178291, −4.96558716576231474715763929001, −4.10607741028612509132773058563, −2.74990832036452731442081790339, −1.8369654961156593909717428877, −0.04046237948435302032344664073, 2.2412402298223499286738415493, 3.14771546793705756463104934990, 4.59898249241994882805011143465, 5.26358710140764535652833962728, 6.49432901291476504351286934782, 7.46186965141975205661282604292, 8.31794688375553556504629936914, 9.67624519541303588482191025506, 10.96334666201489577026073399825, 11.81724799273262713195342068588, 13.04737884350772877427938696147, 13.41470192106093804473626295837, 14.741187586247290075359853384328, 15.41804928109927090723055336599, 16.21205460919008760540161760235, 17.37306051819498949135825371747, 17.99474823641396045801836958175, 19.4967117798390959318195471210, 20.30294348385793304907190849563, 21.35460524576889921765327320458, 21.96499296280817447626713331488, 22.995152500918774742544230756398, 23.67959698487882572358280426167, 24.59069285716959448786366014700, 25.29218196283873896808583653975

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