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Properties

Label 2-2740-2740.539-c0-0-0
Degree $2$
Conductor $2740$
Sign $0.0954 - 0.995i$
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  + (−0.445 + 0.895i)2-s + (−0.602 − 0.798i)4-s + (−0.739 + 0.673i)5-s + (0.982 − 0.183i)8-s + (0.995 + 0.0922i)9-s + (−0.273 − 0.961i)10-s + (−0.111 − 0.252i)13-s + (−0.273 + 0.961i)16-s + (0.361 + 0.0675i)17-s + (−0.526 + 0.850i)18-s + (0.982 + 0.183i)20-s + (0.0922 − 0.995i)25-s + (0.276 + 0.0127i)26-s + (1.59 − 0.890i)29-s + (−0.739 − 0.673i)32-s + ⋯
L(s)  = 1  + (−0.445 + 0.895i)2-s + (−0.602 − 0.798i)4-s + (−0.739 + 0.673i)5-s + (0.982 − 0.183i)8-s + (0.995 + 0.0922i)9-s + (−0.273 − 0.961i)10-s + (−0.111 − 0.252i)13-s + (−0.273 + 0.961i)16-s + (0.361 + 0.0675i)17-s + (−0.526 + 0.850i)18-s + (0.982 + 0.183i)20-s + (0.0922 − 0.995i)25-s + (0.276 + 0.0127i)26-s + (1.59 − 0.890i)29-s + (−0.739 − 0.673i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2740\)    =    \(2^{2} \cdot 5 \cdot 137\)
Sign: $0.0954 - 0.995i$
Analytic conductor: \(1.36743\)
Root analytic conductor: \(1.16937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2740} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2740,\ (\ :0),\ 0.0954 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8661013189\)
\(L(\frac12)\) \(\approx\) \(0.8661013189\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.445 - 0.895i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
137 \( 1 + (-0.932 + 0.361i)T \)
good3 \( 1 + (-0.995 - 0.0922i)T^{2} \)
7 \( 1 + (-0.739 - 0.673i)T^{2} \)
11 \( 1 + (-0.273 + 0.961i)T^{2} \)
13 \( 1 + (0.111 + 0.252i)T + (-0.673 + 0.739i)T^{2} \)
17 \( 1 + (-0.361 - 0.0675i)T + (0.932 + 0.361i)T^{2} \)
19 \( 1 + (0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.526 + 0.850i)T^{2} \)
29 \( 1 + (-1.59 + 0.890i)T + (0.526 - 0.850i)T^{2} \)
31 \( 1 + (0.895 + 0.445i)T^{2} \)
37 \( 1 + 1.79T + T^{2} \)
41 \( 1 + (-1.37 - 1.37i)T + iT^{2} \)
43 \( 1 + (0.895 - 0.445i)T^{2} \)
47 \( 1 + (0.183 - 0.982i)T^{2} \)
53 \( 1 + (-1.60 + 0.377i)T + (0.895 - 0.445i)T^{2} \)
59 \( 1 + (0.982 + 0.183i)T^{2} \)
61 \( 1 + (0.544 - 0.0505i)T + (0.982 - 0.183i)T^{2} \)
67 \( 1 + (-0.673 + 0.739i)T^{2} \)
71 \( 1 + (0.961 - 0.273i)T^{2} \)
73 \( 1 + (-0.719 - 1.85i)T + (-0.739 + 0.673i)T^{2} \)
79 \( 1 + (-0.995 + 0.0922i)T^{2} \)
83 \( 1 + (-0.361 - 0.932i)T^{2} \)
89 \( 1 + (1.89 + 0.634i)T + (0.798 + 0.602i)T^{2} \)
97 \( 1 + (0.195 - 1.40i)T + (-0.961 - 0.273i)T^{2} \)
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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

−9.037448319381469229561521002676, −8.169992854042860176924936709131, −7.69129260949388288076432710457, −6.94994943642986889005816620636, −6.43936303278003866819285537056, −5.43306097013996482685695269384, −4.48811023071595469443626164708, −3.86585150637009422087704301378, −2.57451222332498181510585132572, −1.07367377276971763956520528605, 0.872775799647055023997241217084, 1.89165007860270485235181848335, 3.17239495124675370352435145829, 4.00376537300455700480126522165, 4.61207242710599809333069635464, 5.45044629223982851717191752500, 6.99168912655399373325196558064, 7.35736079272651761351403283006, 8.350073824857094766557296127295, 8.833287187117718634071924733493

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