Properties

Label 2-240-15.2-c1-0-4
Degree $2$
Conductor $240$
Sign $0.920 - 0.391i$
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  + (1.70 − 0.292i)3-s + (−0.707 + 2.12i)5-s + (1 + i)7-s + (2.82 − i)9-s + 1.41i·11-s + (−0.585 + 3.82i)15-s + (1.41 − 1.41i)17-s − 4i·19-s + (2 + 1.41i)21-s + (−2.82 − 2.82i)23-s + (−3.99 − 3i)25-s + (4.53 − 2.53i)27-s − 7.07·29-s + 2·31-s + (0.414 + 2.41i)33-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)3-s + (−0.316 + 0.948i)5-s + (0.377 + 0.377i)7-s + (0.942 − 0.333i)9-s + 0.426i·11-s + (−0.151 + 0.988i)15-s + (0.342 − 0.342i)17-s − 0.917i·19-s + (0.436 + 0.308i)21-s + (−0.589 − 0.589i)23-s + (−0.799 − 0.600i)25-s + (0.872 − 0.487i)27-s − 1.31·29-s + 0.359·31-s + (0.0721 + 0.420i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61316 + 0.328467i\)
\(L(\frac12)\) \(\approx\) \(1.61316 + 0.328467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.70 + 0.292i)T \)
5 \( 1 + (0.707 - 2.12i)T \)
good7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (2.82 + 2.82i)T + 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (5 - 5i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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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

−12.17357955919665420545723235917, −11.27231362804532560212750937811, −10.15896183076687530651832536691, −9.276427151016199512669633371663, −8.138030654360897630184246494722, −7.38028564105894202736701374826, −6.37190787957464485731840504226, −4.64530619211594333345999103072, −3.31221585966140152183897399587, −2.18348590428698455706011520867, 1.61753522619537926076848143016, 3.52772229754449689331233736598, 4.45151706131114117735597423237, 5.79013113471455402614509553785, 7.57450160831616388846090142798, 8.087185093384237320099461870712, 9.086700700392381725424574911571, 9.933370045323478332606324401378, 11.09003167545535410593262849981, 12.22870443400751744384377365327

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