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Properties

Label 2-2760-2760.1739-c0-0-0
Degree $2$
Conductor $2760$
Sign $0.0174 - 0.999i$
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  + (−0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.142 + 0.989i)6-s + (0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s + (0.415 − 0.909i)10-s + (0.415 − 0.909i)12-s + (0.654 − 0.755i)15-s + (−0.654 + 0.755i)16-s + (−0.584 + 0.909i)17-s + (0.654 − 0.755i)18-s + (0.983 + 1.53i)19-s + (−0.841 + 0.540i)20-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.142 + 0.989i)6-s + (0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s + (0.415 − 0.909i)10-s + (0.415 − 0.909i)12-s + (0.654 − 0.755i)15-s + (−0.654 + 0.755i)16-s + (−0.584 + 0.909i)17-s + (0.654 − 0.755i)18-s + (0.983 + 1.53i)19-s + (−0.841 + 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0174 - 0.999i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ 0.0174 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3627088130\)
\(L(\frac12)\) \(\approx\) \(0.3627088130\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
good7 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (0.584 - 0.909i)T + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.983 - 1.53i)T + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (1.37 + 1.19i)T + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + 0.563iT - T^{2} \)
53 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.959 + 0.281i)T^{2} \)
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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

−9.289941025317809695227117630788, −8.246040614230060636895287262696, −7.66005313079464091760620344438, −7.14091458548723721525572608868, −6.14878977616937862409668917896, −5.81472659287168405210790827915, −4.21780110492103056605994336468, −3.34747765944639776521694798604, −2.22764569495316370532409251276, −1.55902875238576897281039814288, 0.33007167565705081706842070996, 1.63816751986699318958397678692, 3.10561128799552443046080469730, 4.50073843639920359042089752159, 5.05624855347130277563332712957, 5.63810196780373441089978307573, 6.57097181770378402653911229847, 7.26980542441221485372405291558, 8.219152230295137133933425967588, 9.066726705460390943379147849357

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