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Properties

Label 2-1340-1340.19-c0-0-0
Degree $2$
Conductor $1340$
Sign $-0.994 - 0.109i$
Analytic cond. $0.668747$
Root an. cond. $0.817769$
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  + (−0.888 − 0.458i)2-s + (−1.11 + 0.326i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (1.13 + 0.219i)6-s + (0.0688 + 1.44i)7-s + (−0.142 − 0.989i)8-s + (0.290 − 0.186i)9-s + (0.235 + 0.971i)10-s + (−0.911 − 0.717i)12-s + (0.601 − 1.31i)14-s + (0.975 + 0.627i)15-s + (−0.327 + 0.945i)16-s + (−0.344 + 0.0328i)18-s + (0.235 − 0.971i)20-s + (−0.549 − 1.58i)21-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (−1.11 + 0.326i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (1.13 + 0.219i)6-s + (0.0688 + 1.44i)7-s + (−0.142 − 0.989i)8-s + (0.290 − 0.186i)9-s + (0.235 + 0.971i)10-s + (−0.911 − 0.717i)12-s + (0.601 − 1.31i)14-s + (0.975 + 0.627i)15-s + (−0.327 + 0.945i)16-s + (−0.344 + 0.0328i)18-s + (0.235 − 0.971i)20-s + (−0.549 − 1.58i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.994 - 0.109i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.994 - 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03463295468\)
\(L(\frac12)\) \(\approx\) \(0.03463295468\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.888 + 0.458i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
good3 \( 1 + (1.11 - 0.326i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \)
11 \( 1 + (-0.928 + 0.371i)T^{2} \)
13 \( 1 + (-0.723 + 0.690i)T^{2} \)
17 \( 1 + (0.327 + 0.945i)T^{2} \)
19 \( 1 + (0.995 + 0.0950i)T^{2} \)
23 \( 1 + (0.723 + 0.690i)T + (0.0475 + 0.998i)T^{2} \)
29 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.723 - 0.690i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.84 + 0.176i)T + (0.981 + 0.189i)T^{2} \)
43 \( 1 + (0.827 + 1.81i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.370 - 1.52i)T + (-0.888 - 0.458i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (1.28 + 0.247i)T + (0.928 + 0.371i)T^{2} \)
71 \( 1 + (0.327 - 0.945i)T^{2} \)
73 \( 1 + (-0.928 - 0.371i)T^{2} \)
79 \( 1 + (-0.235 - 0.971i)T^{2} \)
83 \( 1 + (0.0311 - 0.0899i)T + (-0.786 - 0.618i)T^{2} \)
89 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
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

−10.24443810174594196757224869810, −9.372675315525153989533654632978, −8.588484945020822058396926430819, −8.185020339068151575402181673015, −6.98497888427040330316666183151, −5.99761876823807312376503445428, −5.24461134002119677917123034220, −4.31863165628358947578549288552, −3.05311488894584026205703983499, −1.74815796455683112391524173057, 0.04581148284124836731058522542, 1.47000950432722522779757740882, 3.24119832198307724469508023677, 4.43611339675082576580196153326, 5.52378222074535740978089944021, 6.52040696033630487786890285377, 6.89412463046247675683013063076, 7.67955426898510536320481170726, 8.296964548479033314956582208571, 9.653182886312212892395773191380

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