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Properties

Label 2-3240-360.149-c0-0-5
Degree $2$
Conductor $3240$
Sign $0.342 - 0.939i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s + 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s + 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.865602891\)
\(L(\frac12)\) \(\approx\) \(1.865602891\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.553878682989301208468220561294, −8.146273068801059516560121516847, −7.48306610621289982435207642923, −6.75168984452561472778122541531, −5.78320205913641770868145414206, −5.20105013387499800074080846780, −4.34320634627157855201372412701, −3.61666391580642632152600668482, −2.91420861413574966714953470237, −1.39946721765067327501956454241, 0.966762182452367038959833914728, 2.38658944902249868862699106629, 3.30861555030007517803080849281, 3.74629247742254120842019771332, 4.86588434927753064853923439152, 5.40736776190717007796477326596, 6.39262453117356248693827462754, 7.24428023627683891989971176666, 7.59728141348824773390490575469, 8.759346748622231336065147496119

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