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Properties

Label 1-388-388.7-r0-0-0
Degree $1$
Conductor $388$
Sign $-0.762 + 0.647i$
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  + (0.793 + 0.608i)3-s + (−0.442 + 0.896i)5-s + (−0.997 − 0.0654i)7-s + (0.258 + 0.965i)9-s + (−0.130 + 0.991i)11-s + (0.896 + 0.442i)13-s + (−0.896 + 0.442i)15-s + (−0.0654 − 0.997i)17-s + (−0.555 − 0.831i)19-s + (−0.751 − 0.659i)21-s + (−0.659 + 0.751i)23-s + (−0.608 − 0.793i)25-s + (−0.382 + 0.923i)27-s + (0.321 + 0.946i)29-s + (−0.608 + 0.793i)31-s + ⋯
L(s)  = 1  + (0.793 + 0.608i)3-s + (−0.442 + 0.896i)5-s + (−0.997 − 0.0654i)7-s + (0.258 + 0.965i)9-s + (−0.130 + 0.991i)11-s + (0.896 + 0.442i)13-s + (−0.896 + 0.442i)15-s + (−0.0654 − 0.997i)17-s + (−0.555 − 0.831i)19-s + (−0.751 − 0.659i)21-s + (−0.659 + 0.751i)23-s + (−0.608 − 0.793i)25-s + (−0.382 + 0.923i)27-s + (0.321 + 0.946i)29-s + (−0.608 + 0.793i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $-0.762 + 0.647i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ -0.762 + 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3899117385 + 1.061135088i\)
\(L(\frac12)\) \(\approx\) \(0.3899117385 + 1.061135088i\)
\(L(1)\) \(\approx\) \(0.9053379504 + 0.5501134226i\)
\(L(1)\) \(\approx\) \(0.9053379504 + 0.5501134226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (0.793 + 0.608i)T \)
5 \( 1 + (-0.442 + 0.896i)T \)
7 \( 1 + (-0.997 - 0.0654i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (0.896 + 0.442i)T \)
17 \( 1 + (-0.0654 - 0.997i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (0.321 + 0.946i)T \)
31 \( 1 + (-0.608 + 0.793i)T \)
37 \( 1 + (-0.751 + 0.659i)T \)
41 \( 1 + (-0.946 + 0.321i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (0.659 + 0.751i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (0.946 + 0.321i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.997 - 0.0654i)T \)
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

−24.19356380545035139160556991049, −23.50080500604164600144319152195, −22.566592401098916170966111513965, −21.216209971750323216772121343789, −20.623173459265905026408334783945, −19.57456147565025804286709713071, −19.16465169534216121749769126793, −18.262041951465101246595227547819, −16.966838150451836075110864438486, −16.11901509613889197977110609192, −15.3912451199733098926634115309, −14.22970510045919549886029848510, −13.125112208656141145919255496274, −12.85019482943413992188482746854, −11.82164283341670385678797861289, −10.481185131459318128759175264229, −9.34201857185925877461671392508, −8.3656385794279122224802475200, −8.03531275248915693003549588022, −6.46121785854258391594384209084, −5.80809729189139260473604562493, −3.97505982007847153076911145549, −3.42259043811709951657383995320, −1.97014562774307183020665244739, −0.589980339299139308581901810491, 2.08671400846787782047861890884, 3.16840092341260222865008582033, 3.84871668783955572620998505818, 5.0407608095661200509946667011, 6.68606202186727023834148125407, 7.22343252105392823086935293112, 8.51625126538280600731299755858, 9.46276584781808790783113155805, 10.24963944962898109556705981646, 11.10721682130209414780238361453, 12.30699859585282425967633196183, 13.51626206022093794089903783993, 14.119087670828083251769919243156, 15.37873888583960002794706968239, 15.62124688725334963693539295154, 16.619339626379320150888165785941, 18.06346099186922727492420759007, 18.787509412551775992987802478677, 19.781045722455249240420082985567, 20.2012168710701359540855608928, 21.44617366573105833558665329251, 22.17423191775828376782539389656, 22.986211878452472022329700144896, 23.74323991497328806878849125325, 25.3253117813140504934848020298

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