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Properties

Label 2-3332-3332.1019-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.999 + 0.0213i$
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  + (−0.988 − 0.149i)2-s + (1.36 − 0.930i)3-s + (0.955 + 0.294i)4-s + (−1.48 + 0.716i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.632 − 1.61i)9-s + (−0.722 − 1.84i)11-s + (1.57 − 0.487i)12-s + (−1.23 + 1.54i)13-s + (0.826 + 0.563i)14-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (−0.865 + 1.49i)18-s + (−1.63 + 0.246i)21-s + (0.440 + 1.92i)22-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (1.36 − 0.930i)3-s + (0.955 + 0.294i)4-s + (−1.48 + 0.716i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.632 − 1.61i)9-s + (−0.722 − 1.84i)11-s + (1.57 − 0.487i)12-s + (−1.23 + 1.54i)13-s + (0.826 + 0.563i)14-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (−0.865 + 1.49i)18-s + (−1.63 + 0.246i)21-s + (0.440 + 1.92i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.999 + 0.0213i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.999 + 0.0213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6365970018\)
\(L(\frac12)\) \(\approx\) \(0.6365970018\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
good3 \( 1 + (-1.36 + 0.930i)T + (0.365 - 0.930i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \)
13 \( 1 + (1.23 - 1.54i)T + (-0.222 - 0.974i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.914 - 0.848i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (-0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.603 + 1.53i)T + (-0.733 - 0.680i)T^{2} \)
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

−8.452625942674940758798023324813, −7.68934024666247813350762222416, −7.33526194752282588818220495134, −6.50729787810129263289045536202, −5.89521950536175138939555653031, −4.14394098951933334701633974088, −3.25534866867893562519068498499, −2.58272558800112628394128685031, −1.88249436121175035318284909304, −0.38138126468301668637445155566, 2.24987563009292359872437994132, 2.45987664307698134294671040315, 3.40951926012116440636312238566, 4.53136103298114367708003542646, 5.30888816172932698089532784204, 6.42620800854709266235115683421, 7.30041025606778135741464332554, 7.942895458115921150392549335789, 8.440617595385013351159142691851, 9.262195685670523470235311211918

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