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Properties

Label 1-1368-1368.1139-r1-0-0
Degree $1$
Conductor $1368$
Sign $-0.642 + 0.766i$
Analytic cond. $147.012$
Root an. cond. $147.012$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 35-s + 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 35-s + 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(147.012\)
Root analytic conductor: \(147.012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (1:\ ),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4158655769 - 0.8918266077i\)
\(L(\frac12)\) \(\approx\) \(-0.4158655769 - 0.8918266077i\)
\(L(1)\) \(\approx\) \(0.7713756693 - 0.4615380416i\)
\(L(1)\) \(\approx\) \(0.7713756693 - 0.4615380416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.47800449716979643132196385957, −20.10013374024283052036397846234, −19.736396718181295147335756598215, −18.83249115624790311754875769380, −18.05990059161750873053961017054, −17.66812265492926135052223759455, −16.53163344447482616696886897961, −15.65901525833750581361054875926, −14.96749916687454063316152554257, −14.52270392104097029589277453288, −13.648126739175591252594666526428, −12.467858130530924527467314708696, −11.80095057422049298400143409778, −11.319629761720719524769657831936, −10.33490607437288473718368460595, −9.42480972113514200874018135336, −8.70639860273912062068558817915, −7.713914256530541139515868860979, −6.92346135142092416585810695749, −6.31877005901563365707570821595, −5.065210601342331134453480875059, −4.36327382359309364668048227517, −3.351728527686050486886968830145, −2.29755444505600439967309712385, −1.64210562685548442633627323964, 0.23386375831701832532653213706, 0.80210758007980514261522422372, 1.987246229125660931945203490, 3.26378063851439110789059645255, 4.239674884471773563227311612293, 4.72645520899439805820508455923, 5.82306674846093702448440721941, 6.73694214957814760597411234162, 7.91281000757203474752749686221, 8.1679017210543131577103545974, 9.19085319510835997125573108759, 10.06599082017791341850447806818, 11.09288814536402472814585741415, 11.52562408076072027860282251679, 12.60234007667367181642246276718, 13.206674110204882408343706389646, 14.03229877508594724609100754473, 14.831259953181898514349661356068, 15.732075128104548019549898441586, 16.43521622502016871610799736602, 17.1652492901337241458283405006, 17.66881017151177472738006307599, 18.78977021830569945702800860489, 19.63092445983244161972891120233, 20.2229716567785433290093493551

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