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Properties

Label 2-3332-3332.135-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.788 + 0.615i$
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  + (0.826 − 0.563i)2-s + (1.07 − 0.997i)3-s + (0.365 − 0.930i)4-s + (0.326 − 1.42i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.0858 − 1.14i)9-s + (0.123 + 1.64i)11-s + (−0.535 − 1.36i)12-s + (−1.48 − 0.716i)13-s + (−0.733 − 0.680i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.574 − 0.995i)18-s + (−1.21 − 0.825i)21-s + (1.03 + 1.29i)22-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (1.07 − 0.997i)3-s + (0.365 − 0.930i)4-s + (0.326 − 1.42i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.0858 − 1.14i)9-s + (0.123 + 1.64i)11-s + (−0.535 − 1.36i)12-s + (−1.48 − 0.716i)13-s + (−0.733 − 0.680i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.574 − 0.995i)18-s + (−1.21 − 0.825i)21-s + (1.03 + 1.29i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.643805287\)
\(L(\frac12)\) \(\approx\) \(2.643805287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
good3 \( 1 + (-1.07 + 0.997i)T + (0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (-0.123 - 1.64i)T + (-0.988 + 0.149i)T^{2} \)
13 \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.78 + 0.268i)T + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 + 0.930i)T^{2} \)
53 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \)
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

−8.486055933468817757783081379872, −7.30551017284067067559250722141, −7.12825441457105176537371332551, −6.67677375617829792267924019299, −5.04599577836607542743858719286, −4.71558675284602551113130893236, −3.60168505156113376170457642241, −2.71919204563519525814081229379, −2.16847270106598409535094042219, −1.07231411769789858903287804612, 2.41003159538276845698564349626, 2.85115601639572777960728600911, 3.59415688858260929646012335062, 4.58963459809122223897302548880, 5.04608867991879544415148746046, 6.04426785038665616761933193876, 6.74613164831219358505131455846, 7.67117638926138256578536042968, 8.593087098413169333763051584442, 8.937730197703749650705230046209

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