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Properties

Label 2-3328-13.8-c0-0-3
Degree $2$
Conductor $3328$
Sign $0.881 - 0.471i$
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  + 3-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s + (−1 + i)11-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s − i·17-s + (0.707 − 0.707i)21-s + 1.41i·23-s − 27-s + 1.41·29-s + (−1 + i)33-s + 1.00·35-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + 3-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s + (−1 + i)11-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s − i·17-s + (0.707 − 0.707i)21-s + 1.41i·23-s − 27-s + 1.41·29-s + (−1 + i)33-s + 1.00·35-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.051021658\)
\(L(\frac12)\) \(\approx\) \(2.051021658\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - T + T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−8.815176816241605910503730693584, −8.134712417056477708931251357580, −7.31659805302441499865887836338, −6.97319627322600926168438607803, −5.83505923864179244139873470868, −4.99270317463985505204265935803, −4.14068007307710213243920503090, −3.13427922464015359823109457473, −2.40312227299853544140047820469, −1.59938138577273879508913627901, 1.22605712779122625489598950306, 2.38438478315658059860663845204, 2.91101931258649621932504331457, 4.00521099784402202953118004537, 5.08850249613359022650202838846, 5.68862115361726910998855430883, 6.24246465426955817401799112467, 7.65736983403266536973562258029, 8.379893291577202282182204040294, 8.589300625396816041196966073388

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