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Properties

Label 2-1320-1320.389-c0-0-0
Degree $2$
Conductor $1320$
Sign $-0.999 + 0.0237i$
Analytic cond. $0.658765$
Root an. cond. $0.811643$
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  + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s − 12-s + (0.5 + 0.363i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−1.30 + 0.951i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s − 12-s + (0.5 + 0.363i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−1.30 + 0.951i)17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.999 + 0.0237i$
Analytic conductor: \(0.658765\)
Root analytic conductor: \(0.811643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :0),\ -0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.072529343\)
\(L(\frac12)\) \(\approx\) \(1.072529343\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
good7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−10.70547244624976938242392379021, −9.224063046098773391316719622205, −8.511172244271481060754900892992, −7.78752072085580888503173148090, −6.65878384865882943981720662319, −6.16093654534863976207519699437, −5.03454520629318291687340172928, −4.36216538572637266159846670307, −3.48748954969524568190944767030, −2.74651035365216376872208727295, 0.71278343220170925384294684156, 2.15156971698552441375114783285, 3.10510481720705059222775538238, 4.42753680324677894071124066733, 5.07466080136426688519903076776, 5.91681633080430342356830971110, 7.06819867656289775351181558857, 7.46645439843946306101995306748, 8.657442978896958335157649949801, 9.355095494302129417581334828900

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