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Properties

Label 2-3332-3332.3263-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.926 - 0.375i$
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  + (0.900 + 0.433i)2-s + (−0.433 − 1.90i)3-s + (0.623 + 0.781i)4-s + (0.433 − 1.90i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (−0.781 − 0.376i)11-s + (1.21 − 1.52i)12-s + (−1.62 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (−0.541 − 0.678i)22-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (−0.433 − 1.90i)3-s + (0.623 + 0.781i)4-s + (0.433 − 1.90i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (−0.781 − 0.376i)11-s + (1.21 − 1.52i)12-s + (−1.62 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (−0.541 − 0.678i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.926 - 0.375i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2673660014\)
\(L(\frac12)\) \(\approx\) \(0.2673660014\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + 1.94T + T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.21 + 1.52i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + 0.867T + T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−7.80699470881939644249981268743, −7.38393170057695649668312455791, −7.08531067715683378001581808376, −6.01689171495149065354658698628, −5.54395945562230092680469085991, −5.07336401313689751857624199112, −3.27409030342682316919110011505, −2.80238158492081189414608585623, −1.90841375903739943625478683330, −0.10767528396247352350189559937, 2.34767431397432780283828937594, 3.13828060962826720155907845116, 3.96591024590629382504147936550, 4.48001538875226293895249479961, 5.28461874352438975861129440212, 5.78747084039527516690772350047, 6.68754954247841581002982176960, 7.57032090135059956705884929474, 8.962620039866151314514932569728, 9.571211632777080821384690869910

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