Properties

Label 2-240-5.4-c1-0-5
Degree $2$
Conductor $240$
Sign $-0.447 + 0.894i$
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  i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (−1 + 2i)15-s + 2i·17-s − 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·31-s + 2i·33-s + (−2 + 4i)35-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.43·31-s + 0.348i·33-s + (−0.338 + 0.676i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465077 - 0.752510i\)
\(L(\frac12)\) \(\approx\) \(0.465077 - 0.752510i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2 + i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−11.97204191946259444215266705660, −10.85038935669694538881413320876, −10.13056002929891195176822211936, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.00717100446239630133784520713, −5.61924101980997412333828064393, −4.36151973355833603134652685788, −2.98301548675506377793302167956, −0.72864190327182967399520039641, 2.58342719920451962723501294913, 3.94695391081871525948763714951, 5.02781841842898122069153631094, 6.41014651359295177422336904223, 7.53127668903399757597493950167, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.60982430904786714834563426049, 11.65746295943836746628078584697, 11.97373279112430878810242545265

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