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Properties

Label 2-240-16.13-c1-0-7
Degree $2$
Conductor $240$
Sign $0.819 + 0.572i$
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  + (−0.720 − 1.21i)2-s + (0.707 + 0.707i)3-s + (−0.960 + 1.75i)4-s + (0.707 − 0.707i)5-s + (0.350 − 1.37i)6-s + 0.0588i·7-s + (2.82 − 0.0955i)8-s + 1.00i·9-s + (−1.37 − 0.350i)10-s + (2.23 − 2.23i)11-s + (−1.91 + 0.561i)12-s + (2.84 + 2.84i)13-s + (0.0716 − 0.0424i)14-s + 1.00·15-s + (−2.15 − 3.37i)16-s + 5.98·17-s + ⋯
L(s)  = 1  + (−0.509 − 0.860i)2-s + (0.408 + 0.408i)3-s + (−0.480 + 0.877i)4-s + (0.316 − 0.316i)5-s + (0.143 − 0.559i)6-s + 0.0222i·7-s + (0.999 − 0.0337i)8-s + 0.333i·9-s + (−0.433 − 0.110i)10-s + (0.673 − 0.673i)11-s + (−0.554 + 0.161i)12-s + (0.790 + 0.790i)13-s + (0.0191 − 0.0113i)14-s + 0.258·15-s + (−0.538 − 0.842i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.819 + 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11780 - 0.351562i\)
\(L(\frac12)\) \(\approx\) \(1.11780 - 0.351562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.720 + 1.21i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 0.0588iT - 7T^{2} \)
11 \( 1 + (-2.23 + 2.23i)T - 11iT^{2} \)
13 \( 1 + (-2.84 - 2.84i)T + 13iT^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 + (0.617 + 0.617i)T + 19iT^{2} \)
23 \( 1 + 0.746iT - 23T^{2} \)
29 \( 1 + (1.13 + 1.13i)T + 29iT^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 + (2.01 - 2.01i)T - 37iT^{2} \)
41 \( 1 + 7.71iT - 41T^{2} \)
43 \( 1 + (2.94 - 2.94i)T - 43iT^{2} \)
47 \( 1 + 0.789T + 47T^{2} \)
53 \( 1 + (6.80 - 6.80i)T - 53iT^{2} \)
59 \( 1 + (9.36 - 9.36i)T - 59iT^{2} \)
61 \( 1 + (0.814 + 0.814i)T + 61iT^{2} \)
67 \( 1 + (-5.46 - 5.46i)T + 67iT^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 + (7.55 + 7.55i)T + 83iT^{2} \)
89 \( 1 + 16.3iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.87053847895415562275350867442, −11.01492956807139239830379631064, −10.06753516595100434283319209556, −9.109210643493699523304499260234, −8.609198206374843976207884482495, −7.36686255102227020651488815103, −5.75065891289737852954667152865, −4.22591411264336038700897942593, −3.24374392425295507424912373885, −1.53419147530896542938591305114, 1.53304982238070767114361948645, 3.59609107610099741218273634314, 5.33672836397342844518430372997, 6.32954818874545915373952868818, 7.33729788809254959898363248560, 8.138983243097687762188213907660, 9.231672995855954266195184516581, 10.00774297522443816190978295545, 11.02519546044921491901392898816, 12.43488610148638592678646524512

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