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Properties

Label 2-240-15.8-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.761 - 0.648i$
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  + (0.292 + 1.70i)3-s + (−2.12 − 0.707i)5-s + (−3 + 3i)7-s + (−2.82 + i)9-s + 1.41i·11-s + (0.585 − 3.82i)15-s + (4.24 + 4.24i)17-s − 4i·19-s + (−5.99 − 4.24i)21-s + (−2.82 + 2.82i)23-s + (3.99 + 3i)25-s + (−2.53 − 4.53i)27-s + 1.41·29-s + 2·31-s + (−2.41 + 0.414i)33-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (−0.948 − 0.316i)5-s + (−1.13 + 1.13i)7-s + (−0.942 + 0.333i)9-s + 0.426i·11-s + (0.151 − 0.988i)15-s + (1.02 + 1.02i)17-s − 0.917i·19-s + (−1.30 − 0.925i)21-s + (−0.589 + 0.589i)23-s + (0.799 + 0.600i)25-s + (−0.487 − 0.872i)27-s + 0.262·29-s + 0.359·31-s + (−0.420 + 0.0721i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260794 + 0.708497i\)
\(L(\frac12)\) \(\approx\) \(0.260794 + 0.708497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (2.12 + 0.707i)T \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−12.29750556441347894068584639952, −11.68328191101994391417123197054, −10.45140599430913236298158370391, −9.537005703670381886436002775416, −8.783764413348496783277041921881, −7.81408323593642117313516579540, −6.26127566462899558418680420495, −5.16344354138596553595208317032, −3.90852951428404091091111351875, −2.88494873542836805191514561028, 0.59425860662412881795441285218, 2.98852744497688693170845133413, 3.92757441274037788562580155216, 5.88349383132742390047617470791, 7.00876366820366526098300398911, 7.53002286684444518266838660063, 8.580686441628929254867592533844, 9.944557534987348379410931165911, 10.85249525050472197413425476391, 12.07504244621902666445620978554

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