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Properties

Label 2-3332-476.215-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.0675 - 0.997i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.867 − 1.75i)5-s + (0.382 + 0.923i)8-s + (−0.130 + 0.991i)9-s + (1.75 + 0.867i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.499 − 0.866i)18-s + (−1.92 + 0.382i)20-s + (−1.73 + 2.25i)25-s + (−1.40 − 0.184i)26-s + (−1.08 + 1.63i)29-s + (0.991 − 0.130i)32-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.867 − 1.75i)5-s + (0.382 + 0.923i)8-s + (−0.130 + 0.991i)9-s + (1.75 + 0.867i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.499 − 0.866i)18-s + (−1.92 + 0.382i)20-s + (−1.73 + 2.25i)25-s + (−1.40 − 0.184i)26-s + (−1.08 + 1.63i)29-s + (0.991 − 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.0675 - 0.997i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.0675 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4652835838\)
\(L(\frac12)\) \(\approx\) \(0.4652835838\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 \)
17 \( 1 + (0.991 - 0.130i)T \)
good3 \( 1 + (0.130 - 0.991i)T^{2} \)
5 \( 1 + (0.867 + 1.75i)T + (-0.608 + 0.793i)T^{2} \)
11 \( 1 + (0.793 - 0.608i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (0.258 - 0.965i)T^{2} \)
23 \( 1 + (-0.130 - 0.991i)T^{2} \)
29 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.130 + 0.991i)T^{2} \)
37 \( 1 + (-0.369 - 0.125i)T + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (-0.108 - 1.65i)T + (-0.991 + 0.130i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.389 - 0.0255i)T + (0.991 + 0.130i)T^{2} \)
79 \( 1 + (0.130 + 0.991i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \)
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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

−8.872094825749955388233145043089, −8.456266183963085748170917566319, −7.64195410961032412218022270285, −7.05123276174969463946388078198, −5.96818574326276469281343481282, −5.19403447557829367636331463622, −4.60848183805931868244698130058, −3.81631624266264234253605233900, −2.01359671180098971698926585273, −1.23082012389885787418452093199, 0.39221201625782327861581524558, 2.13531648790975266488702703715, 3.08628477713021158064954517936, 3.55675752436181505720506649426, 4.26069737636854685873755643923, 6.03998636993158365215891971886, 6.52705744392514000385909653456, 7.24821785399673938305573821021, 7.976048401231836002733753088336, 8.487620878812626741875704171432

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