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Properties

Label 2-240-80.3-c1-0-18
Degree $2$
Conductor $240$
Sign $-0.553 + 0.833i$
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  + (−0.481 − 1.32i)2-s + 3-s + (−1.53 + 1.28i)4-s + (−2.17 − 0.539i)5-s + (−0.481 − 1.32i)6-s + (3.00 − 3.00i)7-s + (2.44 + 1.42i)8-s + 9-s + (0.327 + 3.14i)10-s + (−2.91 − 2.91i)11-s + (−1.53 + 1.28i)12-s − 4.96i·13-s + (−5.44 − 2.55i)14-s + (−2.17 − 0.539i)15-s + (0.723 − 3.93i)16-s + (−2.56 + 2.56i)17-s + ⋯
L(s)  = 1  + (−0.340 − 0.940i)2-s + 0.577·3-s + (−0.768 + 0.640i)4-s + (−0.970 − 0.241i)5-s + (−0.196 − 0.542i)6-s + (1.13 − 1.13i)7-s + (0.863 + 0.504i)8-s + 0.333·9-s + (0.103 + 0.994i)10-s + (−0.879 − 0.879i)11-s + (−0.443 + 0.369i)12-s − 1.37i·13-s + (−1.45 − 0.682i)14-s + (−0.560 − 0.139i)15-s + (0.180 − 0.983i)16-s + (−0.622 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.553 + 0.833i$
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: \((2,\ 240,\ (\ :1/2),\ -0.553 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.483542 - 0.901612i\)
\(L(\frac12)\) \(\approx\) \(0.483542 - 0.901612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.481 + 1.32i)T \)
3 \( 1 - T \)
5 \( 1 + (2.17 + 0.539i)T \)
good7 \( 1 + (-3.00 + 3.00i)T - 7iT^{2} \)
11 \( 1 + (2.91 + 2.91i)T + 11iT^{2} \)
13 \( 1 + 4.96iT - 13T^{2} \)
17 \( 1 + (2.56 - 2.56i)T - 17iT^{2} \)
19 \( 1 + (0.174 + 0.174i)T + 19iT^{2} \)
23 \( 1 + (-2.93 - 2.93i)T + 23iT^{2} \)
29 \( 1 + (-4.90 + 4.90i)T - 29iT^{2} \)
31 \( 1 - 5.24iT - 31T^{2} \)
37 \( 1 + 2.27iT - 37T^{2} \)
41 \( 1 - 0.187iT - 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (-0.0810 - 0.0810i)T + 47iT^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (-3.33 + 3.33i)T - 59iT^{2} \)
61 \( 1 + (1.32 + 1.32i)T + 61iT^{2} \)
67 \( 1 - 9.03iT - 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 + (3.50 - 3.50i)T - 73iT^{2} \)
79 \( 1 + 6.75T + 79T^{2} \)
83 \( 1 - 0.203T + 83T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + (-9.90 + 9.90i)T - 97iT^{2} \)
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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

−11.51747206363202475205185328164, −10.84924742183320175873456208513, −10.21523102255181333459225767141, −8.590990431713485930579899675536, −8.110819812934708022544590065729, −7.44674347728048190063306417285, −5.05654669100247555287918993445, −4.05025407353568017567612776040, −2.96223295566897583121723761119, −0.951475304602595604941625985241, 2.25073116764673599137123261887, 4.37049730835532719504846614745, 5.09074771233987256120347452038, 6.80105911640249098777047415196, 7.55892515761869998977656401443, 8.559112901039436649617107244748, 9.029975775690872110520148400888, 10.41107299468980451925901500538, 11.51913082654167850536431340157, 12.42600497551262418746701433623

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