Properties

Label 2-240-80.43-c1-0-17
Degree $2$
Conductor $240$
Sign $0.845 - 0.533i$
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  + (1.32 + 0.481i)2-s + i·3-s + (1.53 + 1.28i)4-s + (0.539 − 2.17i)5-s + (−0.481 + 1.32i)6-s + (3.00 − 3.00i)7-s + (1.42 + 2.44i)8-s − 9-s + (1.76 − 2.62i)10-s + (−2.91 + 2.91i)11-s + (−1.28 + 1.53i)12-s − 4.96·13-s + (5.44 − 2.55i)14-s + (2.17 + 0.539i)15-s + (0.723 + 3.93i)16-s + (−2.56 + 2.56i)17-s + ⋯
L(s)  = 1  + (0.940 + 0.340i)2-s + 0.577i·3-s + (0.768 + 0.640i)4-s + (0.241 − 0.970i)5-s + (−0.196 + 0.542i)6-s + (1.13 − 1.13i)7-s + (0.504 + 0.863i)8-s − 0.333·9-s + (0.557 − 0.830i)10-s + (−0.879 + 0.879i)11-s + (−0.369 + 0.443i)12-s − 1.37·13-s + (1.45 − 0.682i)14-s + (0.560 + 0.139i)15-s + (0.180 + 0.983i)16-s + (−0.622 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09660 + 0.605937i\)
\(L(\frac12)\) \(\approx\) \(2.09660 + 0.605937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.481i)T \)
3 \( 1 - iT \)
5 \( 1 + (-0.539 + 2.17i)T \)
good7 \( 1 + (-3.00 + 3.00i)T - 7iT^{2} \)
11 \( 1 + (2.91 - 2.91i)T - 11iT^{2} \)
13 \( 1 + 4.96T + 13T^{2} \)
17 \( 1 + (2.56 - 2.56i)T - 17iT^{2} \)
19 \( 1 + (-0.174 + 0.174i)T - 19iT^{2} \)
23 \( 1 + (-2.93 - 2.93i)T + 23iT^{2} \)
29 \( 1 + (4.90 + 4.90i)T + 29iT^{2} \)
31 \( 1 + 5.24iT - 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + 0.187iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (0.0810 + 0.0810i)T + 47iT^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (3.33 + 3.33i)T + 59iT^{2} \)
61 \( 1 + (1.32 - 1.32i)T - 61iT^{2} \)
67 \( 1 + 9.03T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 + (-3.50 + 3.50i)T - 73iT^{2} \)
79 \( 1 - 6.75T + 79T^{2} \)
83 \( 1 - 0.203iT - 83T^{2} \)
89 \( 1 + 2.76T + 89T^{2} \)
97 \( 1 + (-9.90 + 9.90i)T - 97iT^{2} \)
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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

−12.41771625527919555771937034188, −11.33999736368327652323198326904, −10.48228366667867476257570675491, −9.366948154219213724531217537699, −7.87112604672488446247086677636, −7.44255541426721368981737523376, −5.65830456128829521305641946172, −4.67152683279549065713206818562, −4.27442166008680489080527117248, −2.16516190987581496939654185171, 2.19965285730942172584003203698, 2.89890597158524400652076911427, 4.95831424552467526840428431974, 5.66856062364382251849222688933, 6.89285610205521164332512382286, 7.80621844486830925110902661761, 9.190914512094331142269915169879, 10.65271036955808951327268568956, 11.21588990119960760378066856139, 12.06710807500846270621116743054

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