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Properties

Label 2-3332-3332.1971-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.801 - 0.598i$
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  + (−0.733 + 0.680i)2-s + (1.95 − 0.294i)3-s + (0.0747 − 0.997i)4-s + (−1.23 + 1.54i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (2.78 − 0.858i)9-s + (−1.40 − 0.432i)11-s + (−0.147 − 1.97i)12-s + (0.326 + 1.42i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−1.45 + 2.52i)18-s + (1.44 + 1.34i)21-s + (1.32 − 0.636i)22-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (1.95 − 0.294i)3-s + (0.0747 − 0.997i)4-s + (−1.23 + 1.54i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (2.78 − 0.858i)9-s + (−1.40 − 0.432i)11-s + (−0.147 − 1.97i)12-s + (0.326 + 1.42i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−1.45 + 2.52i)18-s + (1.44 + 1.34i)21-s + (1.32 − 0.636i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.858968294\)
\(L(\frac12)\) \(\approx\) \(1.858968294\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 - 0.680i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + (-0.826 + 0.563i)T \)
good3 \( 1 + (-1.95 + 0.294i)T + (0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.367 + 0.250i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.88 - 0.582i)T + (0.826 - 0.563i)T^{2} \)
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

−8.568906142170053324462363705657, −8.352184802807978173043066117418, −7.60851346583044570362264333151, −7.08035465911243357182362982162, −6.10067651157915969865619708403, −5.10399599114579954828302882590, −4.23962366134357909752512867211, −3.02421612615896916189864337216, −2.22545697546608737360943963645, −1.54172678863558045964134210776, 1.35612012906003454602534806278, 2.23748452126035914842930989058, 3.05841681744816473345643522186, 3.70606342822045487934250675000, 4.41558787284496910939108645033, 5.51001513911989508036411613555, 7.32497587525810460632487716128, 7.58955916229801554869883680725, 8.105517268152289020637854959102, 8.536747489391904120441557406183

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