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Properties

Label 1-3328-3328.411-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.913 + 0.405i$
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  + (−0.881 + 0.471i)3-s + (−0.773 + 0.634i)5-s + (−0.555 − 0.831i)7-s + (0.555 − 0.831i)9-s + (0.290 + 0.956i)11-s + (0.382 − 0.923i)15-s + (−0.923 + 0.382i)17-s + (0.995 − 0.0980i)19-s + (0.881 + 0.471i)21-s + (0.980 − 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + ⋯
L(s)  = 1  + (−0.881 + 0.471i)3-s + (−0.773 + 0.634i)5-s + (−0.555 − 0.831i)7-s + (0.555 − 0.831i)9-s + (0.290 + 0.956i)11-s + (0.382 − 0.923i)15-s + (−0.923 + 0.382i)17-s + (0.995 − 0.0980i)19-s + (0.881 + 0.471i)21-s + (0.980 − 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.913 + 0.405i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.913 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7700355554 + 0.1632281820i\)
\(L(\frac12)\) \(\approx\) \(0.7700355554 + 0.1632281820i\)
\(L(1)\) \(\approx\) \(0.6552280589 + 0.1234885164i\)
\(L(1)\) \(\approx\) \(0.6552280589 + 0.1234885164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.881 + 0.471i)T \)
5 \( 1 + (-0.773 + 0.634i)T \)
7 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (0.290 + 0.956i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.995 - 0.0980i)T \)
23 \( 1 + (0.980 - 0.195i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.0980 + 0.995i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.881 - 0.471i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.290 - 0.956i)T \)
59 \( 1 + (-0.634 - 0.773i)T \)
61 \( 1 + (-0.471 - 0.881i)T \)
67 \( 1 + (0.881 - 0.471i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (-0.555 + 0.831i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (0.0980 + 0.995i)T \)
89 \( 1 + (0.195 - 0.980i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−18.81120486059719753969034317490, −18.16944258475491030118082142959, −17.413779248318732031817687905869, −16.61814926792983291414641678574, −16.05391692459788670935487076531, −15.69642098550169776119175914659, −14.79707203547190427448842191373, −13.55661268329909094497024965414, −13.22944832233414930229678532486, −12.37178427837945546861315360602, −11.77142466466455456673840448871, −11.38523096556640752822992454493, −10.600468730621673212976272877231, −9.42643723596781045226366115380, −8.94487876976933333685575849049, −8.08218036189032499161725034973, −7.35059736831688711328410265350, −6.529242453326332093763824816237, −5.848009376470897630071252386888, −5.152613695470150085117210356308, −4.449064937383337914174378624033, −3.41355065581472024367419094604, −2.60300829563010909265094497769, −1.3851378880171104094522298717, −0.58213094540500391593427611025, 0.5093912817994910366202381917, 1.6099424734842095023775715777, 3.02054038475574886105543723992, 3.6107360437327496228859065588, 4.47252700857896012171416914698, 4.883356281551866744286417010006, 6.13800027502460947148122296700, 6.87362619866115905438247505244, 7.102554761440155202817692611580, 8.110104182279677150300408099807, 9.268236478452091515423377155436, 9.86765727702310528885812888843, 10.53758018149172181022884567814, 11.19704843889525535652629044082, 11.68950748361927891934328376576, 12.59659737110042558450890111741, 13.117089109043742564336607373933, 14.193067247807245438757381261480, 14.92529731813736232719867051243, 15.5564450749639921712403710138, 16.04404376513660091187814772293, 16.9034441058846207665812885208, 17.3930868650124537156246678482, 18.13650915797045026201553626972, 18.823128767221817095765832272167

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