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Properties

Label 2-240-80.29-c1-0-3
Degree $2$
Conductor $240$
Sign $-0.360 - 0.932i$
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  + (−1.22 − 0.710i)2-s + (−0.707 + 0.707i)3-s + (0.991 + 1.73i)4-s + (−2.15 − 0.607i)5-s + (1.36 − 0.362i)6-s + 2.25·7-s + (0.0216 − 2.82i)8-s − 1.00i·9-s + (2.20 + 2.27i)10-s + (−1.66 + 1.66i)11-s + (−1.92 − 0.527i)12-s + (−4.76 + 4.76i)13-s + (−2.75 − 1.60i)14-s + (1.95 − 1.09i)15-s + (−2.03 + 3.44i)16-s + 6.99i·17-s + ⋯
L(s)  = 1  + (−0.864 − 0.502i)2-s + (−0.408 + 0.408i)3-s + (0.495 + 0.868i)4-s + (−0.962 − 0.271i)5-s + (0.558 − 0.148i)6-s + 0.851·7-s + (0.00765 − 0.999i)8-s − 0.333i·9-s + (0.695 + 0.718i)10-s + (−0.500 + 0.500i)11-s + (−0.556 − 0.152i)12-s + (−1.32 + 1.32i)13-s + (−0.736 − 0.427i)14-s + (0.503 − 0.281i)15-s + (−0.508 + 0.860i)16-s + 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195989 + 0.285964i\)
\(L(\frac12)\) \(\approx\) \(0.195989 + 0.285964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.710i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.15 + 0.607i)T \)
good7 \( 1 - 2.25T + 7T^{2} \)
11 \( 1 + (1.66 - 1.66i)T - 11iT^{2} \)
13 \( 1 + (4.76 - 4.76i)T - 13iT^{2} \)
17 \( 1 - 6.99iT - 17T^{2} \)
19 \( 1 + (2.66 + 2.66i)T + 19iT^{2} \)
23 \( 1 + 4.41T + 23T^{2} \)
29 \( 1 + (-2.59 - 2.59i)T + 29iT^{2} \)
31 \( 1 + 3.93T + 31T^{2} \)
37 \( 1 + (2.01 + 2.01i)T + 37iT^{2} \)
41 \( 1 - 4.50iT - 41T^{2} \)
43 \( 1 + (-7.14 - 7.14i)T + 43iT^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-0.649 - 0.649i)T + 53iT^{2} \)
59 \( 1 + (-5.64 + 5.64i)T - 59iT^{2} \)
61 \( 1 + (5.00 + 5.00i)T + 61iT^{2} \)
67 \( 1 + (-4.95 + 4.95i)T - 67iT^{2} \)
71 \( 1 - 2.33iT - 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 + (5.25 - 5.25i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 4.61iT - 97T^{2} \)
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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

−12.18214114209683209142261759289, −11.32692792928788852806368817317, −10.64524463503991591250890680463, −9.603847885232039132888290896975, −8.544590301458412817641100415198, −7.74933492797815893674482209195, −6.71586101529893587958362381884, −4.79491662123195250084676572974, −3.96965263175045751182214636431, −2.00348661641952596292371807994, 0.37188604612087768446755312961, 2.58510335515447477026617377985, 4.81154583083966854121058103531, 5.77119872824008010009149829065, 7.31366046677542370668944807407, 7.67828604348573151790546270076, 8.560612287662939316383428322670, 10.04685911693640867418857807433, 10.82287821440745210370914835155, 11.67718709403126614521937179742

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