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Properties

Label 2-3332-3332.2311-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.518 + 0.855i$
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  + (−0.623 − 0.781i)2-s + (−0.781 + 0.376i)3-s + (−0.222 + 0.974i)4-s + (0.781 + 0.376i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (−0.153 + 0.193i)9-s + (0.974 + 1.22i)11-s + (−0.193 − 0.846i)12-s + (−0.777 − 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 0.246·18-s + (0.678 + 0.541i)21-s + (0.347 − 1.52i)22-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.781 + 0.376i)3-s + (−0.222 + 0.974i)4-s + (0.781 + 0.376i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (−0.153 + 0.193i)9-s + (0.974 + 1.22i)11-s + (−0.193 − 0.846i)12-s + (−0.777 − 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 0.246·18-s + (0.678 + 0.541i)21-s + (0.347 − 1.52i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4001969771\)
\(L(\frac12)\) \(\approx\) \(0.4001969771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (0.433 + 0.900i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (0.781 - 0.376i)T + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.974 - 1.22i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.347 - 1.52i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + 0.867T + T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 - 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 + 1.56T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
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

−8.786711247073673865564945325519, −7.65094403800409897653895253369, −7.30597840452589895840250397362, −6.48816631054201285406124021005, −5.25949576921756542557536980546, −4.62254955182552635227362329423, −3.84018992718558400269622021308, −2.89634351980615226743397336076, −1.76809461783829333689950856074, −0.37513215123562130472116971132, 1.12726891505975528184913036096, 2.33110968577581435849534891327, 3.68507468604853375877187659551, 4.76854410054671592985907128619, 5.69510665302685502061598089240, 6.20983984980875859929616186666, 6.60932575932823473945285800672, 7.38446880667211691727043155304, 8.641897795333199944262059943857, 8.809420259145874082203289453380

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