Properties

Degree 1
Conductor 73
Sign $-0.764 - 0.644i$
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 + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.422 + 0.906i)5-s + (−0.984 + 0.173i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (−0.906 − 0.422i)11-s + (0.984 + 0.173i)12-s + (0.573 − 0.819i)13-s + (0.819 + 0.573i)14-s + (0.0871 + 0.996i)15-s + (0.173 + 0.984i)16-s + (−0.258 − 0.965i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.422 + 0.906i)5-s + (−0.984 + 0.173i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (−0.906 − 0.422i)11-s + (0.984 + 0.173i)12-s + (0.573 − 0.819i)13-s + (0.819 + 0.573i)14-s + (0.0871 + 0.996i)15-s + (0.173 + 0.984i)16-s + (−0.258 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.764 - 0.644i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.764 - 0.644i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $-0.764 - 0.644i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (60, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (1:\ ),\ -0.764 - 0.644i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2499930368 - 0.6843300242i$
$L(\frac12,\chi)$  $\approx$  $0.2499930368 - 0.6843300242i$
$L(\chi,1)$  $\approx$  0.6382751887 - 0.2945884135i
$L(1,\chi)$  $\approx$  0.6382751887 - 0.2945884135i

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

−31.86213223331703110429396881653, −30.88894119962792802478418170570, −29.040337952177574583213512833888, −28.326220959166391341142766607981, −27.30585639655955379553662944444, −26.11654265654204663609671806651, −25.58278783397507341327997494539, −24.39190668599635962374750068816, −23.29335680868356887298991287345, −21.36827149528098742509492680735, −20.41719092416036415579707830476, −19.49047326621573781019343822678, −18.65340692544314190969037329533, −16.90879591364903435058970946509, −15.88564185927971804071179842522, −15.40501901660167142284048520076, −13.69225209114842040715094005941, −12.277945947821107677169165520475, −10.49722717311771499064178233019, −9.41203603707016588108920455446, −8.55408633901686565915487486856, −7.44422208370135509974782206818, −5.62445693401261183026495179911, −3.78952008572048141695355842980, −1.91789045354134420763306558149, 0.42200986340222041837335847655, 2.692644140105128203332712507276, 3.35118250680391535703325770582, 6.57077904453622257708237560573, 7.462882943041225187641788312224, 8.61437530204960071521713249018, 9.95590223411238057068857406485, 11.01812436080868614572327501391, 12.57564087603752346668849561180, 13.66032550156551798177898933519, 15.34886381542262917850284250829, 16.1313910790353872649665106283, 18.13914886487955627710558430656, 18.533735359905427915288019885347, 19.74001251226555880561681415422, 20.374048053255404913993294827938, 21.89620246772356627365151558477, 23.27724328004232937688405382140, 24.69951900841084051679138029175, 25.95250307026898576232063249103, 26.26191277939519440000396856602, 27.36281409476882277093097674561, 28.93399436611964248490758676565, 29.74598337477741500548890632110, 30.628163482315215806363066478602

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