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Properties

Label 2-240-80.69-c1-0-12
Degree $2$
Conductor $240$
Sign $0.341 + 0.939i$
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  + (−0.550 − 1.30i)2-s + (0.707 + 0.707i)3-s + (−1.39 + 1.43i)4-s + (0.162 − 2.23i)5-s + (0.531 − 1.31i)6-s + 2.93·7-s + (2.63 + 1.02i)8-s + 1.00i·9-s + (−2.99 + 1.01i)10-s + (−0.663 − 0.663i)11-s + (−1.99 + 0.0281i)12-s + (1.12 + 1.12i)13-s + (−1.61 − 3.82i)14-s + (1.69 − 1.46i)15-s + (−0.112 − 3.99i)16-s − 7.47i·17-s + ⋯
L(s)  = 1  + (−0.389 − 0.921i)2-s + (0.408 + 0.408i)3-s + (−0.697 + 0.716i)4-s + (0.0724 − 0.997i)5-s + (0.217 − 0.534i)6-s + 1.10·7-s + (0.931 + 0.363i)8-s + 0.333i·9-s + (−0.946 + 0.321i)10-s + (−0.200 − 0.200i)11-s + (−0.577 + 0.00813i)12-s + (0.312 + 0.312i)13-s + (−0.431 − 1.02i)14-s + (0.436 − 0.377i)15-s + (−0.0281 − 0.999i)16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.341 + 0.939i$
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: \((2,\ 240,\ (\ :1/2),\ 0.341 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994745 - 0.697059i\)
\(L(\frac12)\) \(\approx\) \(0.994745 - 0.697059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.550 + 1.30i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.162 + 2.23i)T \)
good7 \( 1 - 2.93T + 7T^{2} \)
11 \( 1 + (0.663 + 0.663i)T + 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 + 7.47iT - 17T^{2} \)
19 \( 1 + (-0.423 + 0.423i)T - 19iT^{2} \)
23 \( 1 - 6.17T + 23T^{2} \)
29 \( 1 + (2.95 - 2.95i)T - 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (5.53 - 5.53i)T - 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (-0.897 + 0.897i)T - 43iT^{2} \)
47 \( 1 - 4.12iT - 47T^{2} \)
53 \( 1 + (-0.146 + 0.146i)T - 53iT^{2} \)
59 \( 1 + (7.72 + 7.72i)T + 59iT^{2} \)
61 \( 1 + (7.37 - 7.37i)T - 61iT^{2} \)
67 \( 1 + (-8.68 - 8.68i)T + 67iT^{2} \)
71 \( 1 - 8.95iT - 71T^{2} \)
73 \( 1 - 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (9.18 + 9.18i)T + 83iT^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−11.61399875355017512115558722798, −11.20473117920743013008266373920, −9.891150913337552788646462821464, −9.063139618630296542578464590687, −8.416530749993736319254967443198, −7.42576009051811669450074125883, −5.09573728586129739533575968825, −4.57988020374511536861130275970, −2.97908467675940621986125093487, −1.35612082180812258367801196281, 1.83711586504177115011026714741, 3.82880314393169191908544099378, 5.38967191223490458252991788679, 6.45879113390439051943358034935, 7.48421637515606916937713425639, 8.142910516348132702571438273735, 9.115911849813905428916730918088, 10.48812829087621706298669307844, 10.95655142806496549093768124489, 12.49646906257478540580962936448

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