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Properties

Label 2-240-80.3-c1-0-21
Degree $2$
Conductor $240$
Sign $0.150 + 0.988i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.429 − 1.34i)2-s + 3-s + (−1.63 − 1.15i)4-s + (2.23 + 0.0583i)5-s + (0.429 − 1.34i)6-s + (0.747 − 0.747i)7-s + (−2.25 + 1.70i)8-s + 9-s + (1.03 − 2.98i)10-s + (−0.920 − 0.920i)11-s + (−1.63 − 1.15i)12-s + 0.996i·13-s + (−0.686 − 1.32i)14-s + (2.23 + 0.0583i)15-s + (1.32 + 3.77i)16-s + (−0.982 + 0.982i)17-s + ⋯
L(s)  = 1  + (0.303 − 0.952i)2-s + 0.577·3-s + (−0.815 − 0.578i)4-s + (0.999 + 0.0261i)5-s + (0.175 − 0.550i)6-s + (0.282 − 0.282i)7-s + (−0.798 + 0.602i)8-s + 0.333·9-s + (0.328 − 0.944i)10-s + (−0.277 − 0.277i)11-s + (−0.471 − 0.333i)12-s + 0.276i·13-s + (−0.183 − 0.354i)14-s + (0.577 + 0.0150i)15-s + (0.331 + 0.943i)16-s + (−0.238 + 0.238i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.150 + 0.988i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34477 - 1.15585i\)
\(L(\frac12)\) \(\approx\) \(1.34477 - 1.15585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.429 + 1.34i)T \)
3 \( 1 - T \)
5 \( 1 + (-2.23 - 0.0583i)T \)
good7 \( 1 + (-0.747 + 0.747i)T - 7iT^{2} \)
11 \( 1 + (0.920 + 0.920i)T + 11iT^{2} \)
13 \( 1 - 0.996iT - 13T^{2} \)
17 \( 1 + (0.982 - 0.982i)T - 17iT^{2} \)
19 \( 1 + (1.03 + 1.03i)T + 19iT^{2} \)
23 \( 1 + (4.77 + 4.77i)T + 23iT^{2} \)
29 \( 1 + (2.95 - 2.95i)T - 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 - 8.22iT - 37T^{2} \)
41 \( 1 + 5.70iT - 41T^{2} \)
43 \( 1 + 5.22iT - 43T^{2} \)
47 \( 1 + (-0.0548 - 0.0548i)T + 47iT^{2} \)
53 \( 1 - 5.13T + 53T^{2} \)
59 \( 1 + (2.30 - 2.30i)T - 59iT^{2} \)
61 \( 1 + (-10.8 - 10.8i)T + 61iT^{2} \)
67 \( 1 + 8.99iT - 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (-6.35 + 6.35i)T - 73iT^{2} \)
79 \( 1 + 8.76T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + (-1.29 + 1.29i)T - 97iT^{2} \)
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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

−12.03806856816795460585456999131, −10.67488468379544944411475008131, −10.27091674447293709500501662799, −9.099170342363217505227985001526, −8.420746235919704372504173114613, −6.74373717242890843427483437514, −5.47463657717791871777575337605, −4.31594866382038184522833821140, −2.89331882967798078812821476149, −1.68384408308562354372505558581, 2.33153715389657729227335330241, 3.99008743835698268141188672566, 5.32981988532729912802355221539, 6.15396884309050056775445564462, 7.44175590229779130447057905339, 8.276747287834339489852554000734, 9.380198818653838387536077572781, 9.956307895495583526256373973511, 11.54642482418040836216692693098, 12.87819911087422167175123795938

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