Properties

Label 2-240-80.27-c1-0-22
Degree $2$
Conductor $240$
Sign $-0.584 - 0.811i$
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 − 3-s + 2i·4-s + (−1 − 2i)5-s + (1 + i)6-s + (−1 − i)7-s + (2 − 2i)8-s + 9-s + (−1 + 3i)10-s + (−3 + 3i)11-s − 2i·12-s + 4i·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 − 0.577·3-s + i·4-s + (−0.447 − 0.894i)5-s + (0.408 + 0.408i)6-s + (−0.377 − 0.377i)7-s + (0.707 − 0.707i)8-s + 0.333·9-s + (−0.316 + 0.948i)10-s + (−0.904 + 0.904i)11-s − 0.577i·12-s + 1.10i·13-s + 0.534i·14-s + (0.258 + 0.516i)15-s − 16-s + (−1.21 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.584 - 0.811i)\)

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 \( 1 + T \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (3 - 3i)T - 11iT^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
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 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (5 - 5i)T - 47iT^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (5 + 5i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
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

−11.46574759667380122410150261708, −10.59273309588687988988562107263, −9.625350653196612621442010844386, −8.787661221243612341942801285456, −7.62682221860671592295869042249, −6.72848495978425082631292628476, −4.85977465272441622298382063252, −4.02122800468219321565212115666, −2.00613862041050658725312290962, 0, 2.72972912538582356414716305014, 4.68328939645955649737669524800, 6.14015997276311683887837545920, 6.51326708974038905061816688266, 7.944923204248483363061287855152, 8.560454258715767800743701086765, 10.08753467867943356355994031591, 10.73284349028198014132831754272, 11.33281622379642241046405486843

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