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Properties

Label 2-240-80.43-c1-0-5
Degree $2$
Conductor $240$
Sign $0.610 - 0.791i$
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  + (−0.264 − 1.38i)2-s + i·3-s + (−1.85 + 0.735i)4-s + (2 + i)5-s + (1.38 − 0.264i)6-s + (−3.24 + 3.24i)7-s + (1.51 + 2.38i)8-s − 9-s + (0.859 − 3.04i)10-s + (−3.24 + 3.24i)11-s + (−0.735 − 1.85i)12-s + (5.37 + 3.65i)14-s + (−1 + 2i)15-s + (2.91 − 2.73i)16-s + (−0.0586 + 0.0586i)17-s + (0.264 + 1.38i)18-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + 0.577i·3-s + (−0.929 + 0.367i)4-s + (0.894 + 0.447i)5-s + (0.567 − 0.108i)6-s + (−1.22 + 1.22i)7-s + (0.535 + 0.844i)8-s − 0.333·9-s + (0.271 − 0.962i)10-s + (−0.979 + 0.979i)11-s + (−0.212 − 0.536i)12-s + (1.43 + 0.976i)14-s + (−0.258 + 0.516i)15-s + (0.729 − 0.683i)16-s + (−0.0142 + 0.0142i)17-s + (0.0623 + 0.327i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.610 - 0.791i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.610 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819969 + 0.403133i\)
\(L(\frac12)\) \(\approx\) \(0.819969 + 0.403133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 + 1.38i)T \)
3 \( 1 - iT \)
5 \( 1 + (-2 - i)T \)
good7 \( 1 + (3.24 - 3.24i)T - 7iT^{2} \)
11 \( 1 + (3.24 - 3.24i)T - 11iT^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.0586 - 0.0586i)T - 17iT^{2} \)
19 \( 1 + (-4.30 + 4.30i)T - 19iT^{2} \)
23 \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 + 6.49iT - 31T^{2} \)
37 \( 1 - 1.88T + 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + (-6.80 - 6.80i)T + 47iT^{2} \)
53 \( 1 - 9.11iT - 53T^{2} \)
59 \( 1 + (1.36 + 1.36i)T + 59iT^{2} \)
61 \( 1 + (2.05 - 2.05i)T - 61iT^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 + (-1.11 + 1.11i)T - 73iT^{2} \)
79 \( 1 - 8.99T + 79T^{2} \)
83 \( 1 - 4.99iT - 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 + (-9.99 + 9.99i)T - 97iT^{2} \)
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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

−12.26279698097170992571111597148, −11.17586596861366882932826219891, −10.20108641271320765911351071409, −9.459607497990237154100480472309, −9.094551074157665874307145258951, −7.38747780704661821098974353961, −5.83456325091677680283550862036, −4.95550116810802078471031991063, −3.13544126789118907802591041832, −2.42401078885898455519860842618, 0.805116841095489270475009114442, 3.32178592957981652547342159481, 5.06137882150267679840471957123, 6.09540787976338711466223165054, 6.86522730607366141253607047471, 7.916628932453488442544105013763, 8.925165395706399783760666227623, 10.00336053132582665181125222333, 10.57906841046958889362684066205, 12.48263025327702740325038811101

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