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Properties

Label 2-240-240.149-c0-0-1
Degree $2$
Conductor $240$
Sign $0.923 + 0.382i$
Analytic cond. $0.119775$
Root an. cond. $0.346086$
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  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.00·10-s + (−0.707 − 0.707i)12-s − 1.00·15-s − 1.00·16-s + 1.41·17-s + (0.707 + 0.707i)18-s + (−1 + i)19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.00·10-s + (−0.707 − 0.707i)12-s − 1.00·15-s − 1.00·16-s + 1.41·17-s + (0.707 + 0.707i)18-s + (−1 + i)19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.119775\)
Root analytic conductor: \(0.346086\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5917809861\)
\(L(\frac12)\) \(\approx\) \(0.5917809861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.36936163496750345326236536554, −11.42609557929333522777040228548, −9.987440746581778878235120595066, −9.152244553984701345408037980079, −8.043005708260171471716271182966, −7.80746578592877739633446267261, −6.52446969123786007095903519098, −5.31144419502214469638598438828, −3.67167826288751907376634774032, −1.51939273448247247661441585868, 2.53553620062341549250040701672, 3.53036680717894158420722420130, 4.62319019308263877060332591515, 6.77479150238894859714644007434, 7.909413203481590687250027947400, 8.542877462261970093053789337113, 9.682864892071757417871083182559, 10.49354232776174795576261236325, 11.14349063871715647995098067786, 12.19935326316781195879732182893

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