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Properties

Label 2-3328-104.3-c0-0-3
Degree $2$
Conductor $3328$
Sign $0.999 - 0.00641i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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  + i·5-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)29-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)45-s + (−0.5 − 0.866i)49-s + i·53-s + (0.866 + 0.5i)61-s + (0.5 + 0.866i)65-s + 73-s + (−0.499 − 0.866i)81-s + (0.866 + 0.5i)85-s + ⋯
L(s)  = 1  + i·5-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)29-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)45-s + (−0.5 − 0.866i)49-s + i·53-s + (0.866 + 0.5i)61-s + (0.5 + 0.866i)65-s + 73-s + (−0.499 − 0.866i)81-s + (0.866 + 0.5i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.999 - 0.00641i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.999 - 0.00641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.390574239\)
\(L(\frac12)\) \(\approx\) \(1.390574239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.919414470126788049104377318345, −7.943645783486808793948780061088, −7.21060301410169960290821999655, −6.66319597986782397897997951369, −5.90991008490320167257798363236, −5.08790662258467137892836551223, −3.84863458741325057142075658514, −3.38217342727101278275837772047, −2.41610404033864820506153376557, −1.03739681053479753074257913500, 1.23282429344926911721284320887, 2.01485939258307213524789821735, 3.40111890204301787533385152404, 4.29752275679114191870030067396, 4.89141429406323763153044144991, 5.75561460216029227163854048108, 6.47910861268353249226663221548, 7.48668875796155470922828352322, 8.197726071045382444016678235502, 8.631418797162020754593382606086

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