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Properties

Label 2-3328-104.93-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.958 + 0.283i$
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  + (−0.366 − 0.366i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s − 0.732i·25-s + (1.5 − 0.866i)29-s + (1.86 + 0.5i)37-s + (1.86 + 0.5i)41-s + (0.5 − 0.133i)45-s + (−0.866 + 0.5i)49-s − i·53-s + (−0.866 − 0.5i)61-s + (−0.133 + 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s − 0.732i·25-s + (1.5 − 0.866i)29-s + (1.86 + 0.5i)37-s + (1.86 + 0.5i)41-s + (0.5 − 0.133i)45-s + (−0.866 + 0.5i)49-s − i·53-s + (−0.866 − 0.5i)61-s + (−0.133 + 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.958 + 0.283i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.958 + 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133156686\)
\(L(\frac12)\) \(\approx\) \(1.133156686\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−8.496035716560202743467224032704, −7.978129928264267928041697858084, −7.66656119230051686908066232746, −6.37851587140962471785008619214, −5.77454395282607514960973054792, −4.86635912528479448975027052556, −4.29823811408053858463768280750, −3.07440001321462754717056425364, −2.38815573006241071183332501094, −0.892198906343324535952034737952, 1.02724387459941760741150787824, 2.52429185860616203246947830923, 3.25578719646000899301107582398, 4.13542678883333485709505908580, 4.99080894977485011405071949089, 5.95590815885459182680499672331, 6.60064729606784527216270860115, 7.40084114066933685431148503648, 7.954222561041058188154798089519, 9.104241063794921509645928883540

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