Properties

Label 1-388-388.175-r0-0-0
Degree $1$
Conductor $388$
Sign $-0.570 + 0.821i$
Analytic cond. $1.80186$
Root an. cond. $1.80186$
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  + (−0.923 − 0.382i)3-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.831 − 0.555i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.980 − 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (−0.980 + 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.831 − 0.555i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.980 − 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (−0.980 + 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2688565888 + 0.5143908059i\)
\(L(\frac12)\) \(\approx\) \(0.2688565888 + 0.5143908059i\)
\(L(1)\) \(\approx\) \(0.6552173097 + 0.1919048292i\)
\(L(1)\) \(\approx\) \(0.6552173097 + 0.1919048292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
7 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.831 - 0.555i)T \)
17 \( 1 + (-0.831 - 0.555i)T \)
19 \( 1 + (-0.555 + 0.831i)T \)
23 \( 1 + (0.980 - 0.195i)T \)
29 \( 1 + (-0.980 + 0.195i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.195 + 0.980i)T \)
41 \( 1 + (0.195 + 0.980i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (-0.980 - 0.195i)T \)
61 \( 1 + T \)
67 \( 1 + (0.831 + 0.555i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.555 + 0.831i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
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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

−24.260265134502222745951114888283, −23.39241145163791534729367988309, −22.47836347536735234178024939313, −21.59161106290909379516428978927, −20.80528328790693178091641119470, −19.8113567497754450883855009306, −19.114327008919513401359754857158, −17.568726480055054023646558389400, −17.073270524583163744170639827422, −16.534063292768072632961902951966, −15.41028160137781331049286271731, −14.59077651632923693271808613111, −13.25418678662537818103782098972, −12.4430202237785403458719689058, −11.23094597032233743736950229262, −11.12599629180786250300493873386, −9.5666748391543063393390300550, −8.8388025667153338997241643561, −7.47186600003704884132494336112, −6.64704173380419575957932231836, −5.331395492810487727347995871002, −4.39379454807929940686436936796, −3.89460702501632005971470896750, −1.69023872327508107220756256783, −0.39982082936253546724221713370, 1.61166250058357259038952974089, 2.7967094303446408998536642550, 4.32209793368566177725640645571, 5.25680823444241898736045365799, 6.41738790283261829382661694657, 7.145614425122167177059326757953, 8.1199394279185848618807643142, 9.42684478039039213126176761376, 10.642475859662736053089087519692, 11.37118434740584524693290742755, 12.10073867157203128486717902929, 12.85388901829744053682279477200, 14.37930999785587026568787077469, 14.99313961184253026691155144625, 15.91609755767243958826672263322, 17.1018036391614750749473356466, 17.76134759709784476953416167869, 18.63566931272396492599808120349, 19.24673177473486431367458366808, 20.35923944075234055710721331810, 21.69773365732059608740184906992, 22.33119332274791029410806727530, 22.86019864191770726603055251289, 23.87132037293663530469455278393, 24.77450223403287962262437115507

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