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Properties

Label 2-240-80.67-c1-0-23
Degree $2$
Conductor $240$
Sign $-0.330 + 0.943i$
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  + (1.24 − 0.677i)2-s − i·3-s + (1.08 − 1.68i)4-s + (−2.21 − 0.311i)5-s + (−0.677 − 1.24i)6-s + (−1.96 − 1.96i)7-s + (0.202 − 2.82i)8-s − 9-s + (−2.95 + 1.11i)10-s + (0.870 + 0.870i)11-s + (−1.68 − 1.08i)12-s + 5.88·13-s + (−3.77 − 1.10i)14-s + (−0.311 + 2.21i)15-s + (−1.66 − 3.63i)16-s + (2.69 + 2.69i)17-s + ⋯
L(s)  = 1  + (0.877 − 0.479i)2-s − 0.577i·3-s + (0.540 − 0.841i)4-s + (−0.990 − 0.139i)5-s + (−0.276 − 0.506i)6-s + (−0.743 − 0.743i)7-s + (0.0715 − 0.997i)8-s − 0.333·9-s + (−0.935 + 0.352i)10-s + (0.262 + 0.262i)11-s + (−0.485 − 0.312i)12-s + 1.63·13-s + (−1.00 − 0.296i)14-s + (−0.0805 + 0.571i)15-s + (−0.415 − 0.909i)16-s + (0.653 + 0.653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.330 + 0.943i$
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: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971444 - 1.36886i\)
\(L(\frac12)\) \(\approx\) \(0.971444 - 1.36886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.677i)T \)
3 \( 1 + iT \)
5 \( 1 + (2.21 + 0.311i)T \)
good7 \( 1 + (1.96 + 1.96i)T + 7iT^{2} \)
11 \( 1 + (-0.870 - 0.870i)T + 11iT^{2} \)
13 \( 1 - 5.88T + 13T^{2} \)
17 \( 1 + (-2.69 - 2.69i)T + 17iT^{2} \)
19 \( 1 + (-2.40 - 2.40i)T + 19iT^{2} \)
23 \( 1 + (-2.63 + 2.63i)T - 23iT^{2} \)
29 \( 1 + (7.43 - 7.43i)T - 29iT^{2} \)
31 \( 1 + 7.72iT - 31T^{2} \)
37 \( 1 + 4.49T + 37T^{2} \)
41 \( 1 - 4.84iT - 41T^{2} \)
43 \( 1 + 0.461T + 43T^{2} \)
47 \( 1 + (-4.66 + 4.66i)T - 47iT^{2} \)
53 \( 1 - 2.41iT - 53T^{2} \)
59 \( 1 + (6.47 - 6.47i)T - 59iT^{2} \)
61 \( 1 + (-8.50 - 8.50i)T + 61iT^{2} \)
67 \( 1 - 6.40T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (-1.62 - 1.62i)T + 73iT^{2} \)
79 \( 1 - 4.14T + 79T^{2} \)
83 \( 1 - 0.241iT - 83T^{2} \)
89 \( 1 + 2.86T + 89T^{2} \)
97 \( 1 + (-3.18 - 3.18i)T + 97iT^{2} \)
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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

−11.96517823545743020635906445110, −11.11740760876655193009614526757, −10.30947349909355688402224785587, −8.917169692784262532123186041296, −7.59527606813947236760850211670, −6.71668185404612764631156129400, −5.65419815692975901827217134895, −3.99763047230685597752253892297, −3.38363327746588453980612608554, −1.20161317528206236078756743515, 3.14474776610228579319586523367, 3.76176093852254044502467660505, 5.19662571269170820994388088057, 6.17582055377049219230814328968, 7.29957431671961374268265570827, 8.456489367014831697994015250970, 9.292329062536312475471046737249, 10.93445764506584638670533845343, 11.54812510894737399913997913695, 12.39669626549320308755761019072

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