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Properties

Label 2-240-80.67-c1-0-13
Degree $2$
Conductor $240$
Sign $-0.987 - 0.160i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + i·3-s + 2i·4-s + (−2 + i)5-s + (1 − i)6-s + (−1 − i)7-s + (2 − 2i)8-s − 9-s + (3 + i)10-s + (−3 − 3i)11-s − 2·12-s − 4·13-s + 2i·14-s + (−1 − 2i)15-s − 4·16-s + (−5 − 5i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 0.577i·3-s + i·4-s + (−0.894 + 0.447i)5-s + (0.408 − 0.408i)6-s + (−0.377 − 0.377i)7-s + (0.707 − 0.707i)8-s − 0.333·9-s + (0.948 + 0.316i)10-s + (−0.904 − 0.904i)11-s − 0.577·12-s − 1.10·13-s + 0.534i·14-s + (−0.258 − 0.516i)15-s − 16-s + (−1.21 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.987 - 0.160i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 - iT \)
5 \( 1 + (2 - i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + (-5 - 5i)T + 19iT^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (5 - 5i)T - 29iT^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-5 + 5i)T - 59iT^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−11.44070731511314103994309704499, −10.63977517274164255595662155496, −9.884880060055245499857737553362, −8.857047065868751025180901739782, −7.75853070746914982196962350390, −7.02815029088597365226224680588, −5.07276777252595191268308620890, −3.69248331385016832483460079171, −2.81445558198031978654166644256, 0, 2.27787255637531126882853253614, 4.53948994071891593144375627534, 5.62192354131772824561521146262, 7.06796315242512886720949563599, 7.54548964248546985049904035397, 8.615501937392438174984052698640, 9.462852130841844575018306473631, 10.61249216102593455991782692182, 11.69528915933293878403959868774

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