Properties

Label 2-3332-476.3-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.960 - 0.278i$
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 + (1.75 + 0.867i)5-s + (0.382 − 0.923i)8-s + (0.130 + 0.991i)9-s + (−0.867 − 1.75i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (0.499 − 0.866i)18-s + (−0.382 + 1.92i)20-s + (1.73 + 2.25i)25-s + (−1.40 + 0.184i)26-s + (0.324 − 0.216i)29-s + (0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (1.75 + 0.867i)5-s + (0.382 − 0.923i)8-s + (0.130 + 0.991i)9-s + (−0.867 − 1.75i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (0.499 − 0.866i)18-s + (−0.382 + 1.92i)20-s + (1.73 + 2.25i)25-s + (−1.40 + 0.184i)26-s + (0.324 − 0.216i)29-s + (0.991 + 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.960 - 0.278i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.960 - 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273371854\)
\(L(\frac12)\) \(\approx\) \(1.273371854\)
\(L(1)\) 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 \( 1 + (0.130 - 0.991i)T \)
good3 \( 1 + (-0.130 - 0.991i)T^{2} \)
5 \( 1 + (-1.75 - 0.867i)T + (0.608 + 0.793i)T^{2} \)
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 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.130 + 0.991i)T^{2} \)
37 \( 1 + (0.630 + 1.85i)T + (-0.793 + 0.608i)T^{2} \)
41 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
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 \( 1 + (1.65 + 0.108i)T + (0.991 + 0.130i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.0255 - 0.389i)T + (-0.991 + 0.130i)T^{2} \)
79 \( 1 + (-0.130 + 0.991i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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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

−8.906572927938275455003150703106, −8.290369545618799268835511393530, −7.43731779483543622526429218893, −6.65389886199971352569235196625, −5.91817923838878924253267644525, −5.24178976411179511259768267158, −3.84627038869678246989392452206, −2.95987400926581643371621419588, −2.13742003438485068584642975382, −1.49894270594565298247226418717, 1.14387686549834291822959227425, 1.72176382044842997479479630411, 2.95052226107287757435291145743, 4.52302895273101632404562567057, 5.12239656079059723432725366479, 6.12743925811774558846464245599, 6.36546266231503704343649967903, 7.10340121461616930569113981426, 8.373367251606871957573746233311, 8.919368041174497734755873374798

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