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Properties

Label 2-240-16.13-c1-0-8
Degree $2$
Conductor $240$
Sign $0.996 + 0.0819i$
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  + (−0.167 + 1.40i)2-s + (−0.707 − 0.707i)3-s + (−1.94 − 0.470i)4-s + (0.707 − 0.707i)5-s + (1.11 − 0.874i)6-s − 1.41i·7-s + (0.985 − 2.65i)8-s + 1.00i·9-s + (0.874 + 1.11i)10-s + (4.22 − 4.22i)11-s + (1.04 + 1.70i)12-s + (1.33 + 1.33i)13-s + (1.98 + 0.236i)14-s − 1.00·15-s + (3.55 + 1.82i)16-s − 1.14·17-s + ⋯
L(s)  = 1  + (−0.118 + 0.992i)2-s + (−0.408 − 0.408i)3-s + (−0.971 − 0.235i)4-s + (0.316 − 0.316i)5-s + (0.453 − 0.357i)6-s − 0.534i·7-s + (0.348 − 0.937i)8-s + 0.333i·9-s + (0.276 + 0.351i)10-s + (1.27 − 1.27i)11-s + (0.300 + 0.492i)12-s + (0.370 + 0.370i)13-s + (0.530 + 0.0632i)14-s − 0.258·15-s + (0.889 + 0.457i)16-s − 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.996 + 0.0819i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.996 + 0.0819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00674 - 0.0413199i\)
\(L(\frac12)\) \(\approx\) \(1.00674 - 0.0413199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.167 - 1.40i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + (-4.22 + 4.22i)T - 11iT^{2} \)
13 \( 1 + (-1.33 - 1.33i)T + 13iT^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 + (-1.05 - 1.05i)T + 19iT^{2} \)
23 \( 1 + 7.77iT - 23T^{2} \)
29 \( 1 + (-2.94 - 2.94i)T + 29iT^{2} \)
31 \( 1 - 0.389T + 31T^{2} \)
37 \( 1 + (4.28 - 4.28i)T - 37iT^{2} \)
41 \( 1 + 5.45iT - 41T^{2} \)
43 \( 1 + (4.91 - 4.91i)T - 43iT^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (0.863 - 0.863i)T - 53iT^{2} \)
59 \( 1 + (-8.50 + 8.50i)T - 59iT^{2} \)
61 \( 1 + (-2.22 - 2.22i)T + 61iT^{2} \)
67 \( 1 + (-2.94 - 2.94i)T + 67iT^{2} \)
71 \( 1 - 3.27iT - 71T^{2} \)
73 \( 1 + 1.84iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (-11.1 - 11.1i)T + 83iT^{2} \)
89 \( 1 - 18.6iT - 89T^{2} \)
97 \( 1 + 5.44T + 97T^{2} \)
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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

−12.25979218760865503457540135228, −11.14226780920927745435357097236, −10.06833170725086743302329817372, −8.858754319649384493881674986067, −8.262292567164014871370387069087, −6.73799644480141087256639858055, −6.35648526885832925106153039190, −5.07544390668155707244342745066, −3.82253194480850436544569627569, −1.04220694679017133731360942740, 1.77264149504675336110008633775, 3.40943797691900513414197140946, 4.60094642590227954152834485294, 5.76948239911116316052384954943, 7.13596787706467351218286484953, 8.666823349954901354450155286636, 9.581156617734002365740323752626, 10.12209060378303417700689493262, 11.38304221930953251667231518280, 11.84160010067880912134957517990

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