Properties

Label 2-240-20.3-c1-0-1
Degree $2$
Conductor $240$
Sign $0.793 - 0.608i$
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  + (−0.707 + 0.707i)3-s + (2.22 + 0.224i)5-s + (0.317 + 0.317i)7-s − 1.00i·9-s + 1.41i·11-s + (2.44 + 2.44i)13-s + (−1.73 + 1.41i)15-s + (−0.449 + 0.449i)17-s + 6.29·19-s − 0.449·21-s + (−4.87 + 4.87i)23-s + (4.89 + i)25-s + (0.707 + 0.707i)27-s − 5.34i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.994 + 0.100i)5-s + (0.120 + 0.120i)7-s − 0.333i·9-s + 0.426i·11-s + (0.679 + 0.679i)13-s + (−0.447 + 0.365i)15-s + (−0.109 + 0.109i)17-s + 1.44·19-s − 0.0980·21-s + (−1.01 + 1.01i)23-s + (0.979 + 0.200i)25-s + (0.136 + 0.136i)27-s − 0.993i·29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22899 + 0.417356i\)
\(L(\frac12)\) \(\approx\) \(1.22899 + 0.417356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-2.22 - 0.224i)T \)
good7 \( 1 + (-0.317 - 0.317i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + (0.449 - 0.449i)T - 17iT^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 + (4.87 - 4.87i)T - 23iT^{2} \)
29 \( 1 + 5.34iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + (6.44 - 6.44i)T - 37iT^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + (-6.29 + 6.29i)T - 43iT^{2} \)
47 \( 1 + (8.34 + 8.34i)T + 47iT^{2} \)
53 \( 1 + (2 + 2i)T + 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + (-0.635 - 0.635i)T + 67iT^{2} \)
71 \( 1 - 1.27iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 4.09T + 79T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - 83iT^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−11.95301964627212477744839914842, −11.39811336785794407205559839296, −10.02922072244405860125069596159, −9.687219047514281974711147926930, −8.472024460358868642056796997467, −7.04992601705968311591090175986, −5.98337766585449843441420157296, −5.12146012154965946890971268222, −3.67990078789532933993168205903, −1.85924119390877560639412830447, 1.37484720226801905958451630843, 3.12002058493849750662825070748, 5.00692857403998576349037879401, 5.88148364475025713589300536145, 6.84423069496962298958575007876, 8.108975387994200193022926307656, 9.120464500720444259758478829847, 10.28737005839941527815425014777, 10.94797876536633699872132365300, 12.16751772038577914333851423431

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