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Properties

Label 2-2740-2740.1187-c0-0-0
Degree $2$
Conductor $2740$
Sign $0.218 - 0.975i$
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  + (0.850 + 0.526i)2-s + (0.445 + 0.895i)4-s + (−0.638 − 0.769i)5-s + (−0.0922 + 0.995i)8-s + (0.0461 + 0.998i)9-s + (−0.138 − 0.990i)10-s + (−0.135 − 0.0285i)13-s + (−0.602 + 0.798i)16-s + (1.53 − 0.141i)17-s + (−0.486 + 0.873i)18-s + (0.403 − 0.914i)20-s + (−0.183 + 0.982i)25-s + (−0.100 − 0.0956i)26-s + (0.848 + 1.44i)29-s + (−0.932 + 0.361i)32-s + ⋯
L(s)  = 1  + (0.850 + 0.526i)2-s + (0.445 + 0.895i)4-s + (−0.638 − 0.769i)5-s + (−0.0922 + 0.995i)8-s + (0.0461 + 0.998i)9-s + (−0.138 − 0.990i)10-s + (−0.135 − 0.0285i)13-s + (−0.602 + 0.798i)16-s + (1.53 − 0.141i)17-s + (−0.486 + 0.873i)18-s + (0.403 − 0.914i)20-s + (−0.183 + 0.982i)25-s + (−0.100 − 0.0956i)26-s + (0.848 + 1.44i)29-s + (−0.932 + 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2740\)    =    \(2^{2} \cdot 5 \cdot 137\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(1.36743\)
Root analytic conductor: \(1.16937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2740} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2740,\ (\ :0),\ 0.218 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.815507680\)
\(L(\frac12)\) \(\approx\) \(1.815507680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (0.638 + 0.769i)T \)
137 \( 1 + (-0.183 + 0.982i)T \)
good3 \( 1 + (-0.0461 - 0.998i)T^{2} \)
7 \( 1 + (-0.932 + 0.361i)T^{2} \)
11 \( 1 + (-0.798 - 0.602i)T^{2} \)
13 \( 1 + (0.135 + 0.0285i)T + (0.914 + 0.403i)T^{2} \)
17 \( 1 + (-1.53 + 0.141i)T + (0.982 - 0.183i)T^{2} \)
19 \( 1 + (-0.526 - 0.850i)T^{2} \)
23 \( 1 + (0.486 - 0.873i)T^{2} \)
29 \( 1 + (-0.848 - 1.44i)T + (-0.486 + 0.873i)T^{2} \)
31 \( 1 + (-0.973 + 0.228i)T^{2} \)
37 \( 1 - 1.94iT - T^{2} \)
41 \( 1 + (-0.475 + 1.14i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.973 + 0.228i)T^{2} \)
47 \( 1 + (0.769 + 0.638i)T^{2} \)
53 \( 1 + (1.09 + 1.38i)T + (-0.228 + 0.973i)T^{2} \)
59 \( 1 + (0.0922 + 0.995i)T^{2} \)
61 \( 1 + (1.59 + 0.0737i)T + (0.995 + 0.0922i)T^{2} \)
67 \( 1 + (0.403 - 0.914i)T^{2} \)
71 \( 1 + (-0.990 - 0.138i)T^{2} \)
73 \( 1 + (-0.0762 + 0.0521i)T + (0.361 - 0.932i)T^{2} \)
79 \( 1 + (-0.0461 + 0.998i)T^{2} \)
83 \( 1 + (0.565 + 0.824i)T^{2} \)
89 \( 1 + (-1.17 - 0.844i)T + (0.317 + 0.948i)T^{2} \)
97 \( 1 + (0.127 + 1.84i)T + (-0.990 + 0.138i)T^{2} \)
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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

−8.807488750929528892824860654728, −8.213868484913890074758222909509, −7.63043774410147298854290722912, −6.99820327795726844807286330080, −5.91359726613735943564762976486, −5.04973347826002267176699044673, −4.78450531840084533853226610651, −3.66735281311615077240494578386, −2.92170916013151191433677985777, −1.54403174451968869812835449427, 0.969909730416675131386674352162, 2.47127590384476211304332988141, 3.26850876411389134550743539967, 3.91844703123597245388227121010, 4.69590180340158047605376197414, 6.00983908430596000495082771551, 6.17714437050636946834187442000, 7.33829948241207894269099187885, 7.77818025847351406874561936488, 9.063851712175111580317309159920

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