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Properties

Label 2-240-20.7-c1-0-2
Degree $2$
Conductor $240$
Sign $0.755 - 0.655i$
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 + (−0.224 + 2.22i)5-s + (3.14 − 3.14i)7-s + 1.00i·9-s + 1.41i·11-s + (−2.44 + 2.44i)13-s + (−1.73 + 1.41i)15-s + (4.44 + 4.44i)17-s + 0.635·19-s + 4.44·21-s + (−2.04 − 2.04i)23-s + (−4.89 − i)25-s + (−0.707 + 0.707i)27-s − 9.34i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.100 + 0.994i)5-s + (1.18 − 1.18i)7-s + 0.333i·9-s + 0.426i·11-s + (−0.679 + 0.679i)13-s + (−0.447 + 0.365i)15-s + (1.07 + 1.07i)17-s + 0.145·19-s + 0.970·21-s + (−0.427 − 0.427i)23-s + (−0.979 − 0.200i)25-s + (−0.136 + 0.136i)27-s − 1.73i·29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39745 + 0.521778i\)
\(L(\frac12)\) \(\approx\) \(1.39745 + 0.521778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.224 - 2.22i)T \)
good7 \( 1 + (-3.14 + 3.14i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (-4.44 - 4.44i)T + 17iT^{2} \)
19 \( 1 - 0.635T + 19T^{2} \)
23 \( 1 + (2.04 + 2.04i)T + 23iT^{2} \)
29 \( 1 + 9.34iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + (1.55 + 1.55i)T + 37iT^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 + (-0.635 - 0.635i)T + 43iT^{2} \)
47 \( 1 + (5.51 - 5.51i)T - 47iT^{2} \)
53 \( 1 + (2 - 2i)T - 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 - 2.89T + 61T^{2} \)
67 \( 1 + (-6.29 + 6.29i)T - 67iT^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 + (-3 + 3i)T - 73iT^{2} \)
79 \( 1 + 9.75T + 79T^{2} \)
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

−12.00826853064248297358948858295, −11.13120194669358276169269147170, −10.31621845858811008141961663756, −9.635450802630028945945385763678, −7.88002059106949841207212988015, −7.65714861364846184345796169038, −6.25132644272921907285217418742, −4.60063221497993749331731080442, −3.77666414638299488091156137477, −2.06823761243242344176880823045, 1.49839896983758720461681348397, 3.07657551701670098253775551160, 5.03144286266106297850108938134, 5.47401791763435416639709600300, 7.30789553522961502156117431552, 8.277684820066252016880554996352, 8.807660006706194385423232137884, 9.896503003125242626890691256140, 11.40037309222205723866332371863, 12.15428845150764390156830472063

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