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Properties

Label 2-240-16.13-c1-0-13
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: $\chi_{240} (61, \cdot )$
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

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

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