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Properties

Label 1-3328-3328.1109-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.378 + 0.925i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
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  + (−0.986 − 0.162i)3-s + (0.956 − 0.290i)5-s + (−0.659 + 0.751i)7-s + (0.946 + 0.321i)9-s + (−0.412 − 0.910i)11-s + (−0.991 + 0.130i)15-s + (0.991 + 0.130i)17-s + (−0.0327 + 0.999i)19-s + (0.773 − 0.634i)21-s + (0.997 + 0.0654i)23-s + (0.831 − 0.555i)25-s + (−0.881 − 0.471i)27-s + (−0.812 + 0.582i)29-s + (0.707 + 0.707i)31-s + (0.258 + 0.965i)33-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)3-s + (0.956 − 0.290i)5-s + (−0.659 + 0.751i)7-s + (0.946 + 0.321i)9-s + (−0.412 − 0.910i)11-s + (−0.991 + 0.130i)15-s + (0.991 + 0.130i)17-s + (−0.0327 + 0.999i)19-s + (0.773 − 0.634i)21-s + (0.997 + 0.0654i)23-s + (0.831 − 0.555i)25-s + (−0.881 − 0.471i)27-s + (−0.812 + 0.582i)29-s + (0.707 + 0.707i)31-s + (0.258 + 0.965i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9394032483 + 0.6308481926i\)
\(L(\frac12)\) \(\approx\) \(0.9394032483 + 0.6308481926i\)
\(L(1)\) \(\approx\) \(0.8669930461 + 0.07052907392i\)
\(L(1)\) \(\approx\) \(0.8669930461 + 0.07052907392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.986 - 0.162i)T \)
5 \( 1 + (0.956 - 0.290i)T \)
7 \( 1 + (-0.659 + 0.751i)T \)
11 \( 1 + (-0.412 - 0.910i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.0327 + 0.999i)T \)
23 \( 1 + (0.997 + 0.0654i)T \)
29 \( 1 + (-0.812 + 0.582i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.999 + 0.0327i)T \)
41 \( 1 + (-0.0654 + 0.997i)T \)
43 \( 1 + (0.986 - 0.162i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.0980 + 0.995i)T \)
59 \( 1 + (-0.973 - 0.227i)T \)
61 \( 1 + (0.935 - 0.352i)T \)
67 \( 1 + (-0.162 + 0.986i)T \)
71 \( 1 + (-0.751 - 0.659i)T \)
73 \( 1 + (0.980 - 0.195i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.471 + 0.881i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
97 \( 1 + (0.965 - 0.258i)T \)
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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

−18.74917243850238384758567067419, −17.71546806965726042702263137044, −17.378400553833482400969761243214, −16.854398215634741950408504316619, −16.07593984888357143371896485746, −15.33846237025346767198199047173, −14.64158989037601194076103499554, −13.634344363471422169851053040412, −13.11968183596986837665545642709, −12.55537732892346142750439090138, −11.67132249062811414505734829575, −10.78677223892164500396399146228, −10.34773437136549860085986134561, −9.653357278511129385778489555228, −9.24409478845849913153462731778, −7.76117570302663205306857559080, −6.994523038523750812673187550326, −6.64507463555853411339121396845, −5.67128133221046346626909839156, −5.1248102079681918320597715350, −4.31300102442523591947577018328, −3.36261168076286832124774177345, −2.42319316730726097218803717219, −1.42660556100848991756838899005, −0.4461799011891529952846891285, 0.997070395918716088771385661490, 1.68284486620909685970947863627, 2.80893950778257899475431533975, 3.51914990182951634870844363824, 4.861724909899491410878616734139, 5.43147525629261102351453075359, 5.97468611783668257977516507369, 6.4831202435149626773513176059, 7.47269375610385985681418874223, 8.418486870536686665074956611178, 9.18586551415209392544752105936, 9.94185584755832335833854153555, 10.492459431572438484995449686933, 11.251046362224264372220676626369, 12.18928971125882130354870311828, 12.62921658352327341495911679340, 13.2260964112685903036232415793, 14.00242738317773035412267947221, 14.82326104446224873412335913016, 15.83346243183996914788031270331, 16.34358981731763637257758612449, 16.86385307228149348686090975451, 17.534034342264509755500120775277, 18.378813160295256217973399842201, 18.80415726315176448337171090894

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