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Properties

Label 2-3332-3332.1495-c0-0-2
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  + (−0.955 + 0.294i)2-s + (−0.680 − 1.73i)3-s + (0.826 − 0.563i)4-s + (1.16 + 1.45i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−1.80 + 1.67i)9-s + (−0.432 − 0.400i)11-s + (−1.53 − 1.04i)12-s + (0.425 − 1.86i)13-s + (0.930 + 0.365i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (1.23 − 2.13i)18-s + (−0.548 + 1.77i)21-s + (0.531 + 0.255i)22-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (−0.680 − 1.73i)3-s + (0.826 − 0.563i)4-s + (1.16 + 1.45i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−1.80 + 1.67i)9-s + (−0.432 − 0.400i)11-s + (−1.53 − 1.04i)12-s + (0.425 − 1.86i)13-s + (0.930 + 0.365i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (1.23 − 2.13i)18-s + (−0.548 + 1.77i)21-s + (0.531 + 0.255i)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: $\chi_{3332} (1495, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4230543702\)
\(L(\frac12)\) \(\approx\) \(0.4230543702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (0.781 + 0.623i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
good3 \( 1 + (0.680 + 1.73i)T + (-0.733 + 0.680i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (0.432 + 0.400i)T + (0.0747 + 0.997i)T^{2} \)
13 \( 1 + (-0.425 + 1.86i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.145 + 1.94i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.781 + 1.35i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.268 - 0.129i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (-0.680 - 1.17i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.535 + 0.496i)T + (0.0747 - 0.997i)T^{2} \)
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

−7.987888046473916228717754805081, −7.74798283267776786327087067249, −6.89058780995390188991204613401, −6.33597643708578859228812208938, −5.80794559920882278711925542433, −4.95676512041816027529702951235, −2.98569468219038885480339049238, −2.54070694411978905098580204868, −0.969150968131458083727713899063, −0.47456260015251222676412502069, 1.73946102456305619138633222701, 3.09338640828374147787118999156, 3.67857328106320680022660249644, 4.56071996690779469970604172663, 5.48612304857675647512506754350, 6.35714123667791602897867017405, 6.79208929937042037806693690105, 8.142341410820305595842093582201, 8.949471433196710333813880849562, 9.326258080441165893110557405239

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