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Properties

Label 2-240-80.69-c1-0-5
Degree $2$
Conductor $240$
Sign $0.257 - 0.966i$
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  + (−1.39 + 0.258i)2-s + (0.707 + 0.707i)3-s + (1.86 − 0.719i)4-s + (0.404 + 2.19i)5-s + (−1.16 − 0.800i)6-s + 1.81·7-s + (−2.40 + 1.48i)8-s + 1.00i·9-s + (−1.13 − 2.95i)10-s + (−0.331 − 0.331i)11-s + (1.82 + 0.810i)12-s + (−0.0310 − 0.0310i)13-s + (−2.52 + 0.469i)14-s + (−1.26 + 1.84i)15-s + (2.96 − 2.68i)16-s + 1.00i·17-s + ⋯
L(s)  = 1  + (−0.983 + 0.183i)2-s + (0.408 + 0.408i)3-s + (0.933 − 0.359i)4-s + (0.180 + 0.983i)5-s + (−0.476 − 0.326i)6-s + 0.686·7-s + (−0.851 + 0.524i)8-s + 0.333i·9-s + (−0.357 − 0.933i)10-s + (−0.100 − 0.100i)11-s + (0.527 + 0.233i)12-s + (−0.00860 − 0.00860i)13-s + (−0.674 + 0.125i)14-s + (−0.327 + 0.475i)15-s + (0.740 − 0.671i)16-s + 0.243i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.257 - 0.966i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786016 + 0.604168i\)
\(L(\frac12)\) \(\approx\) \(0.786016 + 0.604168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.258i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.404 - 2.19i)T \)
good7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 + (0.331 + 0.331i)T + 11iT^{2} \)
13 \( 1 + (0.0310 + 0.0310i)T + 13iT^{2} \)
17 \( 1 - 1.00iT - 17T^{2} \)
19 \( 1 + (2.08 - 2.08i)T - 19iT^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + (6.28 - 6.28i)T - 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (0.0723 - 0.0723i)T - 37iT^{2} \)
41 \( 1 + 3.06iT - 41T^{2} \)
43 \( 1 + (-3.78 + 3.78i)T - 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (-7.04 + 7.04i)T - 53iT^{2} \)
59 \( 1 + (6.68 + 6.68i)T + 59iT^{2} \)
61 \( 1 + (-2.89 + 2.89i)T - 61iT^{2} \)
67 \( 1 + (0.150 + 0.150i)T + 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (-5.48 - 5.48i)T + 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−11.94709295177454438100154880528, −10.89675660006498237783557334501, −10.50117789069885048065955735804, −9.402448091826456533088602520352, −8.466813748042824173902417696225, −7.53081310155724321326770616874, −6.56939982228008863681184409762, −5.26522923576574337729487212983, −3.39142432659236940974105938997, −2.02500846014107328012895778151, 1.17180515667186208380048204019, 2.58144907407042171533325302707, 4.48202575580082726764731411455, 5.98819357287602495440328181247, 7.33219569345678545997813758564, 8.155877761496883281301632162843, 8.959019536717981767016837754648, 9.712656903887029566183288309878, 11.00098579112317154520615688841, 11.81546451760620624698926145998

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