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Properties

Label 2-2760-2760.1259-c0-0-3
Degree $2$
Conductor $2760$
Sign $-0.904 - 0.427i$
Analytic cond. $1.37741$
Root an. cond. $1.17363$
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  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)3-s + (−0.959 − 0.281i)4-s + (0.841 − 0.540i)5-s + (−0.415 + 0.909i)6-s + (−0.415 + 0.909i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)10-s + (0.841 + 0.540i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (−1.14 − 0.989i)17-s + (0.654 − 0.755i)18-s + (−1.49 + 1.29i)19-s + (−0.959 + 0.281i)20-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)3-s + (−0.959 − 0.281i)4-s + (0.841 − 0.540i)5-s + (−0.415 + 0.909i)6-s + (−0.415 + 0.909i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)10-s + (0.841 + 0.540i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (−1.14 − 0.989i)17-s + (0.654 − 0.755i)18-s + (−1.49 + 1.29i)19-s + (−0.959 + 0.281i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.904 - 0.427i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ -0.904 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5882878191\)
\(L(\frac12)\) \(\approx\) \(0.5882878191\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good7 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (1.49 - 1.29i)T + (0.142 - 0.989i)T^{2} \)
29 \( 1 + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (0.841 + 0.540i)T^{2} \)
47 \( 1 + 1.81iT - T^{2} \)
53 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.415 - 0.909i)T^{2} \)
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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

−8.697188264197949371543728656256, −8.056920994483679702051773727112, −6.75387315904262593270149471525, −6.07724606483325454644780102944, −5.40397015237743841073159748996, −4.57835432853612896563862236726, −3.99412784301769547739987731637, −2.31377701752415702490354328244, −1.86101544470968252159147319459, −0.39245696745847917768444291421, 1.68790998473676547073395566366, 3.17766377458667681165487409575, 4.30923936265539741291423412224, 4.87081161981972494267114516464, 5.80833391346569544521866457571, 6.39484327233523509733046350588, 6.78965275204037229556778557202, 7.66638595170267855289204523691, 8.894436037142627932545505108296, 9.170351121002156295971864292837

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