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Properties

Label 2-3332-3332.1495-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  + (0.955 − 0.294i)2-s + (−0.266 − 0.680i)3-s + (0.826 − 0.563i)4-s + (−0.455 − 0.571i)6-s + (−0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (1.40 + 1.29i)11-s + (−0.603 − 0.411i)12-s + (−0.425 + 1.86i)13-s + (−0.365 + 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (1.72 + 0.829i)22-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (−0.266 − 0.680i)3-s + (0.826 − 0.563i)4-s + (−0.455 − 0.571i)6-s + (−0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (1.40 + 1.29i)11-s + (−0.603 − 0.411i)12-s + (−0.425 + 1.86i)13-s + (−0.365 + 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (1.72 + 0.829i)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

\(L(\frac{1}{2})\) \(\approx\) \(2.246547469\)
\(L(\frac12)\) \(\approx\) \(2.246547469\)
\(L(1)\) 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 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
good3 \( 1 + (0.266 + 0.680i)T + (-0.733 + 0.680i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \)
13 \( 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0332 - 0.443i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
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 \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.199205246674514842318658224276, −7.46662315394007968908305174012, −6.95552046449477963964985644840, −6.53631098396123687330796528636, −5.87591839597914848311735663058, −4.62411971589807727501542588801, −4.30788202330039600581508142773, −3.15226638043441795505611865683, −2.09130813610060044664796470791, −1.44193082633587682056472683209, 1.22566605082284894282357981892, 2.97362964193008432063377579915, 3.48867612891534832571942982020, 4.23612500687160718951848652774, 5.01209024679849794518841880692, 5.85277508068631038198497304899, 6.44163803049818955634231352863, 7.17901306241707183224877304353, 8.098321046637619834752668418104, 8.669418904240607130225814318028

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