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Properties

Label 2-2760-2760.2309-c0-0-0
Degree $2$
Conductor $2760$
Sign $-0.899 - 0.436i$
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  + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (−0.273 − 0.0801i)11-s + (0.959 + 0.281i)12-s + (−0.345 − 0.755i)13-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (−1.10 − 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (−0.273 − 0.0801i)11-s + (0.959 + 0.281i)12-s + (−0.345 − 0.755i)13-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (−1.10 − 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.899 - 0.436i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ -0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7794936154\)
\(L(\frac12)\) \(\approx\) \(0.7794936154\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good7 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + 1.30T + T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 + 0.989i)T^{2} \)
67 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
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

−9.450293389369342470890059301413, −8.482673575717504454280154651049, −8.050778830576275016080444959143, −7.07747449787553320314450533178, −6.65274266901521433803773423005, −5.44976026556937811865485834294, −4.82296976081455325146438162566, −3.26200252978250307153904431918, −2.89024348520881505944264407003, −1.80190063864815655431380576533, 0.58169815597296767272780588489, 2.01116765203578090380005690230, 2.29109919046232169043889908200, 3.75556189775399679952431032829, 4.53151817825248969889044171094, 5.97240781871529613613070085962, 6.55709359506781174962152529596, 7.43273209443876669035045518449, 8.128184506010303035891355218066, 8.664118344016942496441680951961

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