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Properties

Label 2-240-16.5-c1-0-14
Degree $2$
Conductor $240$
Sign $-0.988 - 0.148i$
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  + (−0.0861 − 1.41i)2-s + (−0.707 + 0.707i)3-s + (−1.98 + 0.243i)4-s + (−0.707 − 0.707i)5-s + (1.05 + 0.937i)6-s − 2.76i·7-s + (0.514 + 2.78i)8-s − 1.00i·9-s + (−0.937 + 1.05i)10-s + (−3.51 − 3.51i)11-s + (1.23 − 1.57i)12-s + (−4.55 + 4.55i)13-s + (−3.90 + 0.238i)14-s + 1.00·15-s + (3.88 − 0.965i)16-s − 5.00·17-s + ⋯
L(s)  = 1  + (−0.0609 − 0.998i)2-s + (−0.408 + 0.408i)3-s + (−0.992 + 0.121i)4-s + (−0.316 − 0.316i)5-s + (0.432 + 0.382i)6-s − 1.04i·7-s + (0.181 + 0.983i)8-s − 0.333i·9-s + (−0.296 + 0.334i)10-s + (−1.05 − 1.05i)11-s + (0.355 − 0.454i)12-s + (−1.26 + 1.26i)13-s + (−1.04 + 0.0636i)14-s + 0.258·15-s + (0.970 − 0.241i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.988 - 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0318736 + 0.427588i\)
\(L(\frac12)\) \(\approx\) \(0.0318736 + 0.427588i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0861 + 1.41i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 2.76iT - 7T^{2} \)
11 \( 1 + (3.51 + 3.51i)T + 11iT^{2} \)
13 \( 1 + (4.55 - 4.55i)T - 13iT^{2} \)
17 \( 1 + 5.00T + 17T^{2} \)
19 \( 1 + (0.812 - 0.812i)T - 19iT^{2} \)
23 \( 1 + 7.48iT - 23T^{2} \)
29 \( 1 + (-6.03 + 6.03i)T - 29iT^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + (-1.08 - 1.08i)T + 37iT^{2} \)
41 \( 1 - 3.15iT - 41T^{2} \)
43 \( 1 + (3.10 + 3.10i)T + 43iT^{2} \)
47 \( 1 + 2.76T + 47T^{2} \)
53 \( 1 + (-6.41 - 6.41i)T + 53iT^{2} \)
59 \( 1 + (5.13 + 5.13i)T + 59iT^{2} \)
61 \( 1 + (2.49 - 2.49i)T - 61iT^{2} \)
67 \( 1 + (3.14 - 3.14i)T - 67iT^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 8.95T + 79T^{2} \)
83 \( 1 + (2.86 - 2.86i)T - 83iT^{2} \)
89 \( 1 + 7.23iT - 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−11.55294866720836643094375719567, −10.64558551226251568931451737197, −10.07576452692768081356701318671, −8.882690825816589349978833277384, −7.948761693918946352417097854777, −6.48119868431914056568607896359, −4.73402191187320886116458541912, −4.30796750328523193249285736428, −2.63208756239835800612948500241, −0.35732841634674972553154137508, 2.65457124946551799849302899390, 4.80086726052005073263233130238, 5.45339603976749663414145689066, 6.74580197121315235929870697877, 7.56685359285304517539428911636, 8.405587323654594366500376820915, 9.677149498972041566044072288312, 10.53506236324616092148302864752, 11.97826674692526613114967883539, 12.68514943131995138560768006296

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