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Properties

Label 2-3332-476.279-c0-0-1
Degree $2$
Conductor $3332$
Sign $-0.135 - 0.990i$
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  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (1.08 − 0.216i)5-s + (0.923 − 0.382i)8-s + (0.382 + 0.923i)9-s + (−0.216 + 1.08i)10-s + (1 + i)13-s + i·16-s + (−0.382 + 0.923i)17-s − 18-s + (−0.923 − 0.617i)20-s + (0.216 − 0.0897i)25-s + (−1.30 + 0.541i)26-s + (−1.63 + 0.324i)29-s + (−0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (1.08 − 0.216i)5-s + (0.923 − 0.382i)8-s + (0.382 + 0.923i)9-s + (−0.216 + 1.08i)10-s + (1 + i)13-s + i·16-s + (−0.382 + 0.923i)17-s − 18-s + (−0.923 − 0.617i)20-s + (0.216 − 0.0897i)25-s + (−1.30 + 0.541i)26-s + (−1.63 + 0.324i)29-s + (−0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.135 - 0.990i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.135 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227661670\)
\(L(\frac12)\) \(\approx\) \(1.227661670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 \)
17 \( 1 + (0.382 - 0.923i)T \)
good3 \( 1 + (-0.382 - 0.923i)T^{2} \)
5 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
97 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
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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

−8.980068915202811715868581538401, −8.330783998170380150620377610558, −7.46042468403722050466587981776, −6.79367709706006810692297921905, −5.88501089487096725604937127772, −5.60871292832768869792417810431, −4.53294544040660625011824602240, −3.86580807294393818615755398860, −2.05971571441448430910075645070, −1.52519391527340899580865129749, 0.909859101426476994500068359206, 1.94429569771089455934424791282, 2.91933398413382598881907258715, 3.64220783691536117804935198785, 4.59872219267987249139177859934, 5.64509022129563463739862496777, 6.24079212784694546754272915972, 7.24884497719123227907869547385, 7.996412376623181727365598802433, 8.987183303320061800956201414169

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