Properties

Label 2-240-80.29-c1-0-18
Degree $2$
Conductor $240$
Sign $0.563 + 0.826i$
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  + (−0.456 + 1.33i)2-s + (0.707 − 0.707i)3-s + (−1.58 − 1.22i)4-s + (−1.50 − 1.65i)5-s + (0.623 + 1.26i)6-s − 2.58·7-s + (2.35 − 1.55i)8-s − 1.00i·9-s + (2.90 − 1.25i)10-s + (4.39 − 4.39i)11-s + (−1.98 + 0.254i)12-s + (−0.417 + 0.417i)13-s + (1.18 − 3.46i)14-s + (−2.23 − 0.110i)15-s + (1.00 + 3.87i)16-s − 4.40i·17-s + ⋯
L(s)  = 1  + (−0.323 + 0.946i)2-s + (0.408 − 0.408i)3-s + (−0.791 − 0.611i)4-s + (−0.671 − 0.741i)5-s + (0.254 + 0.518i)6-s − 0.978·7-s + (0.834 − 0.551i)8-s − 0.333i·9-s + (0.918 − 0.395i)10-s + (1.32 − 1.32i)11-s + (−0.572 + 0.0734i)12-s + (−0.115 + 0.115i)13-s + (0.316 − 0.926i)14-s + (−0.576 − 0.0286i)15-s + (0.252 + 0.967i)16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.563 + 0.826i$
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: \((2,\ 240,\ (\ :1/2),\ 0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716955 - 0.379040i\)
\(L(\frac12)\) \(\approx\) \(0.716955 - 0.379040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 - 1.33i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (1.50 + 1.65i)T \)
good7 \( 1 + 2.58T + 7T^{2} \)
11 \( 1 + (-4.39 + 4.39i)T - 11iT^{2} \)
13 \( 1 + (0.417 - 0.417i)T - 13iT^{2} \)
17 \( 1 + 4.40iT - 17T^{2} \)
19 \( 1 + (4.53 + 4.53i)T + 19iT^{2} \)
23 \( 1 + 0.281T + 23T^{2} \)
29 \( 1 + (-3.73 - 3.73i)T + 29iT^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + (-5.26 - 5.26i)T + 37iT^{2} \)
41 \( 1 - 5.16iT - 41T^{2} \)
43 \( 1 + (-2.66 - 2.66i)T + 43iT^{2} \)
47 \( 1 + 7.45iT - 47T^{2} \)
53 \( 1 + (-2.89 - 2.89i)T + 53iT^{2} \)
59 \( 1 + (4.60 - 4.60i)T - 59iT^{2} \)
61 \( 1 + (0.211 + 0.211i)T + 61iT^{2} \)
67 \( 1 + (-7.17 + 7.17i)T - 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + (-4.56 + 4.56i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 6.78iT - 97T^{2} \)
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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

−12.14492850436294897448768288217, −11.05028062252650333185936759534, −9.416332246094212436985599812576, −8.993233545281960518368360337776, −8.122560944717573061788216467123, −6.90639144125351870468599375507, −6.23709422659525933150675808851, −4.71280795157079031701521629158, −3.43964224141699094321091854873, −0.72476655195163809963581580531, 2.22234052887567904627261395996, 3.72141758760095844161416353999, 4.18005816002601969519214869621, 6.36444109792255052970443334033, 7.52999826162623309115043540000, 8.612580690284722259535051153910, 9.682861212770318812217822202329, 10.22889614166267989961995094036, 11.20038784335195152957701940570, 12.36072032203765928160322279548

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