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Properties

Label 2-2740-2740.139-c0-0-1
Degree $2$
Conductor $2740$
Sign $-0.974 + 0.224i$
Analytic cond. $1.36743$
Root an. cond. $1.16937$
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  + (−0.739 − 0.673i)2-s + (0.0922 + 0.995i)4-s + (0.850 − 0.526i)5-s + (0.602 − 0.798i)8-s + (−0.895 − 0.445i)9-s + (−0.982 − 0.183i)10-s + (−0.748 − 1.34i)13-s + (−0.982 + 0.183i)16-s + (−0.961 − 1.27i)17-s + (0.361 + 0.932i)18-s + (0.602 + 0.798i)20-s + (0.445 − 0.895i)25-s + (−0.352 + 1.49i)26-s + (−0.987 + 1.44i)29-s + (0.850 + 0.526i)32-s + ⋯
L(s)  = 1  + (−0.739 − 0.673i)2-s + (0.0922 + 0.995i)4-s + (0.850 − 0.526i)5-s + (0.602 − 0.798i)8-s + (−0.895 − 0.445i)9-s + (−0.982 − 0.183i)10-s + (−0.748 − 1.34i)13-s + (−0.982 + 0.183i)16-s + (−0.961 − 1.27i)17-s + (0.361 + 0.932i)18-s + (0.602 + 0.798i)20-s + (0.445 − 0.895i)25-s + (−0.352 + 1.49i)26-s + (−0.987 + 1.44i)29-s + (0.850 + 0.526i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2740\)    =    \(2^{2} \cdot 5 \cdot 137\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(1.36743\)
Root analytic conductor: \(1.16937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2740} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2740,\ (\ :0),\ -0.974 + 0.224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5521935713\)
\(L(\frac12)\) \(\approx\) \(0.5521935713\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
137 \( 1 + (0.273 + 0.961i)T \)
good3 \( 1 + (0.895 + 0.445i)T^{2} \)
7 \( 1 + (0.850 + 0.526i)T^{2} \)
11 \( 1 + (-0.982 + 0.183i)T^{2} \)
13 \( 1 + (0.748 + 1.34i)T + (-0.526 + 0.850i)T^{2} \)
17 \( 1 + (0.961 + 1.27i)T + (-0.273 + 0.961i)T^{2} \)
19 \( 1 + (0.739 - 0.673i)T^{2} \)
23 \( 1 + (0.361 + 0.932i)T^{2} \)
29 \( 1 + (0.987 - 1.44i)T + (-0.361 - 0.932i)T^{2} \)
31 \( 1 + (0.673 - 0.739i)T^{2} \)
37 \( 1 + 1.34T + T^{2} \)
41 \( 1 + (1.29 - 1.29i)T - iT^{2} \)
43 \( 1 + (0.673 + 0.739i)T^{2} \)
47 \( 1 + (-0.798 + 0.602i)T^{2} \)
53 \( 1 + (-0.252 - 0.111i)T + (0.673 + 0.739i)T^{2} \)
59 \( 1 + (0.602 + 0.798i)T^{2} \)
61 \( 1 + (-1.75 + 0.876i)T + (0.602 - 0.798i)T^{2} \)
67 \( 1 + (-0.526 + 0.850i)T^{2} \)
71 \( 1 + (-0.183 + 0.982i)T^{2} \)
73 \( 1 + (-1.72 + 0.489i)T + (0.850 - 0.526i)T^{2} \)
79 \( 1 + (0.895 - 0.445i)T^{2} \)
83 \( 1 + (0.961 - 0.273i)T^{2} \)
89 \( 1 + (-0.457 - 0.0211i)T + (0.995 + 0.0922i)T^{2} \)
97 \( 1 + (1.08 + 0.903i)T + (0.183 + 0.982i)T^{2} \)
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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

−8.696479907494129180100884970003, −8.293641358203314094778362563175, −7.20870818017978039413951856466, −6.55204141871647917855095376123, −5.33778921992439147285182660924, −4.91846469544663244904542256627, −3.43498608685423159261405017646, −2.77827446683018131623514520040, −1.81918811057896654903989122229, −0.40924197753604612541840023947, 1.94624834191975084559641424732, 2.25791584051169835499219167467, 3.87570949086655654243716682326, 5.02712773181937324016697890677, 5.68378940745050665986251414423, 6.51014099260813765912561881478, 6.92325525485946577968073996425, 7.897463749831360181697098500516, 8.692934273450556880314811562354, 9.207748850192465024254124539821

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