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Properties

Label 2-240-80.29-c1-0-8
Degree $2$
Conductor $240$
Sign $0.553 - 0.832i$
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  + (−0.382 + 1.36i)2-s + (−0.707 + 0.707i)3-s + (−1.70 − 1.04i)4-s + (1.38 − 1.75i)5-s + (−0.692 − 1.23i)6-s + 4.66·7-s + (2.07 − 1.92i)8-s − 1.00i·9-s + (1.85 + 2.56i)10-s + (1.23 − 1.23i)11-s + (1.94 − 0.471i)12-s + (−4.12 + 4.12i)13-s + (−1.78 + 6.34i)14-s + (0.258 + 2.22i)15-s + (1.83 + 3.55i)16-s − 3.20i·17-s + ⋯
L(s)  = 1  + (−0.270 + 0.962i)2-s + (−0.408 + 0.408i)3-s + (−0.853 − 0.520i)4-s + (0.620 − 0.784i)5-s + (−0.282 − 0.503i)6-s + 1.76·7-s + (0.731 − 0.681i)8-s − 0.333i·9-s + (0.587 + 0.809i)10-s + (0.372 − 0.372i)11-s + (0.561 − 0.136i)12-s + (−1.14 + 1.14i)13-s + (−0.476 + 1.69i)14-s + (0.0666 + 0.573i)15-s + (0.458 + 0.888i)16-s − 0.778i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996666 + 0.534384i\)
\(L(\frac12)\) \(\approx\) \(0.996666 + 0.534384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 1.36i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.38 + 1.75i)T \)
good7 \( 1 - 4.66T + 7T^{2} \)
11 \( 1 + (-1.23 + 1.23i)T - 11iT^{2} \)
13 \( 1 + (4.12 - 4.12i)T - 13iT^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 + (-3.73 - 3.73i)T + 19iT^{2} \)
23 \( 1 - 0.714T + 23T^{2} \)
29 \( 1 + (1.24 + 1.24i)T + 29iT^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (-2.33 - 2.33i)T + 37iT^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 + (1.31 + 1.31i)T + 43iT^{2} \)
47 \( 1 + 1.18iT - 47T^{2} \)
53 \( 1 + (9.35 + 9.35i)T + 53iT^{2} \)
59 \( 1 + (6.22 - 6.22i)T - 59iT^{2} \)
61 \( 1 + (4.44 + 4.44i)T + 61iT^{2} \)
67 \( 1 + (6.37 - 6.37i)T - 67iT^{2} \)
71 \( 1 + 6.23iT - 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (-4.88 + 4.88i)T - 83iT^{2} \)
89 \( 1 + 2.20iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−12.09060821354136493088830161274, −11.40127449024951616767162586148, −10.00897724343356159193982636737, −9.309758805153547211147380896963, −8.356240014899532039815992922830, −7.39591213189189599191266040382, −6.03487795567199014613260904893, −4.95214950173511511874402984297, −4.54981598532963429388383677013, −1.46999472936586566724280349334, 1.53655481315017471780665431530, 2.74649086196028714431935254628, 4.63124981852305667371331868341, 5.54436133585041643136198206180, 7.29985449668091521874880512766, 7.970763053244801713478468887441, 9.296385574235283877830117217647, 10.40308799973943152795020625974, 10.98327917550995447144363072294, 11.82046791195025715828811078688

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