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Properties

Label 2-3328-13.5-c0-0-0
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: $\chi_{3328} (3073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4057717202\)
\(L(\frac12)\) \(\approx\) \(0.4057717202\)
\(L(1)\) 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

−9.183102011137444634977294392921, −8.077692401077722799556148721052, −7.83062905611901218089442144434, −7.10588432961513753404328750591, −6.25874252835575784352321788199, −5.37877938037877617628355175986, −4.13982285254917035511581341742, −3.36062727917648475904459819597, −3.08818343251953014960460426997, −1.87572452092590075395201543127, 0.18988098491569389653830170019, 2.28483093510862661854618010448, 2.69126668323538374050197051887, 3.64420775745917578899784562946, 4.70055820479314528143632738219, 5.23582222920935900080502249867, 6.25255651128486167551986674601, 7.41052505523729724094255638181, 7.78686871874897934100994343943, 8.446803772448472072337516189676

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