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Properties

Label 2-240-16.5-c1-0-9
Degree $2$
Conductor $240$
Sign $0.900 - 0.434i$
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  + (1.18 + 0.768i)2-s + (0.707 − 0.707i)3-s + (0.817 + 1.82i)4-s + (0.707 + 0.707i)5-s + (1.38 − 0.295i)6-s − 4.92i·7-s + (−0.432 + 2.79i)8-s − 1.00i·9-s + (0.295 + 1.38i)10-s + (2.45 + 2.45i)11-s + (1.86 + 0.712i)12-s + (−2.93 + 2.93i)13-s + (3.78 − 5.84i)14-s + 1.00·15-s + (−2.66 + 2.98i)16-s − 5.77·17-s + ⋯
L(s)  = 1  + (0.839 + 0.543i)2-s + (0.408 − 0.408i)3-s + (0.408 + 0.912i)4-s + (0.316 + 0.316i)5-s + (0.564 − 0.120i)6-s − 1.86i·7-s + (−0.152 + 0.988i)8-s − 0.333i·9-s + (0.0935 + 0.437i)10-s + (0.741 + 0.741i)11-s + (0.539 + 0.205i)12-s + (−0.812 + 0.812i)13-s + (1.01 − 1.56i)14-s + 0.258·15-s + (−0.665 + 0.746i)16-s − 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13249 + 0.488057i\)
\(L(\frac12)\) \(\approx\) \(2.13249 + 0.488057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 0.768i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 4.92iT - 7T^{2} \)
11 \( 1 + (-2.45 - 2.45i)T + 11iT^{2} \)
13 \( 1 + (2.93 - 2.93i)T - 13iT^{2} \)
17 \( 1 + 5.77T + 17T^{2} \)
19 \( 1 + (0.984 - 0.984i)T - 19iT^{2} \)
23 \( 1 + 0.539iT - 23T^{2} \)
29 \( 1 + (-6.81 + 6.81i)T - 29iT^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + (6.00 + 6.00i)T + 37iT^{2} \)
41 \( 1 - 5.17iT - 41T^{2} \)
43 \( 1 + (0.180 + 0.180i)T + 43iT^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 + (0.146 + 0.146i)T + 53iT^{2} \)
59 \( 1 + (-3.13 - 3.13i)T + 59iT^{2} \)
61 \( 1 + (1.87 - 1.87i)T - 61iT^{2} \)
67 \( 1 + (-8.02 + 8.02i)T - 67iT^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 - 7.71T + 79T^{2} \)
83 \( 1 + (1.62 - 1.62i)T - 83iT^{2} \)
89 \( 1 - 9.54iT - 89T^{2} \)
97 \( 1 - 4.39T + 97T^{2} \)
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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

−12.46541464192613463858569450230, −11.43903652925959335554676311872, −10.37075681128398656589280092250, −9.202747605013103880883510542288, −7.82127458894245781667319126127, −6.92029594050111917060940734669, −6.57201942468462261880996003229, −4.58505848146294683469003560421, −3.90902116366023369746317868366, −2.16433917989421995390101164614, 2.17039248600833582738008452887, 3.17611203791938402297219212367, 4.78929853960774771597655607351, 5.57292679380061147327622180827, 6.64082018757141753699754185514, 8.646721133982817769741872268779, 9.083597712350143488150766443688, 10.23000501522900433747336772477, 11.30392706977987512322081624123, 12.19027703953097754506019773300

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