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Properties

Label 2-240-80.69-c1-0-19
Degree $2$
Conductor $240$
Sign $-0.929 + 0.368i$
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  + (−1.36 − 0.386i)2-s + (0.707 + 0.707i)3-s + (1.70 + 1.05i)4-s + (−1.98 − 1.03i)5-s + (−0.688 − 1.23i)6-s − 3.91·7-s + (−1.90 − 2.08i)8-s + 1.00i·9-s + (2.30 + 2.16i)10-s + (−2.93 − 2.93i)11-s + (0.459 + 1.94i)12-s + (−0.732 − 0.732i)13-s + (5.33 + 1.51i)14-s + (−0.674 − 2.13i)15-s + (1.78 + 3.57i)16-s − 2.89i·17-s + ⋯
L(s)  = 1  + (−0.961 − 0.273i)2-s + (0.408 + 0.408i)3-s + (0.850 + 0.525i)4-s + (−0.887 − 0.461i)5-s + (−0.281 − 0.504i)6-s − 1.48·7-s + (−0.674 − 0.738i)8-s + 0.333i·9-s + (0.727 + 0.686i)10-s + (−0.884 − 0.884i)11-s + (0.132 + 0.561i)12-s + (−0.203 − 0.203i)13-s + (1.42 + 0.404i)14-s + (−0.174 − 0.550i)15-s + (0.447 + 0.894i)16-s − 0.701i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.929 + 0.368i$
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: \((2,\ 240,\ (\ :1/2),\ -0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0347329 - 0.182018i\)
\(L(\frac12)\) \(\approx\) \(0.0347329 - 0.182018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.386i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.98 + 1.03i)T \)
good7 \( 1 + 3.91T + 7T^{2} \)
11 \( 1 + (2.93 + 2.93i)T + 11iT^{2} \)
13 \( 1 + (0.732 + 0.732i)T + 13iT^{2} \)
17 \( 1 + 2.89iT - 17T^{2} \)
19 \( 1 + (-1.67 + 1.67i)T - 19iT^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 + (4.99 - 4.99i)T - 29iT^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (-6.41 + 6.41i)T - 37iT^{2} \)
41 \( 1 + 0.00577iT - 41T^{2} \)
43 \( 1 + (2.23 - 2.23i)T - 43iT^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (-5.55 + 5.55i)T - 53iT^{2} \)
59 \( 1 + (-3.83 - 3.83i)T + 59iT^{2} \)
61 \( 1 + (-9.30 + 9.30i)T - 61iT^{2} \)
67 \( 1 + (3.85 + 3.85i)T + 67iT^{2} \)
71 \( 1 + 1.15iT - 71T^{2} \)
73 \( 1 + 7.98T + 73T^{2} \)
79 \( 1 - 0.843T + 79T^{2} \)
83 \( 1 + (5.20 + 5.20i)T + 83iT^{2} \)
89 \( 1 - 5.40iT - 89T^{2} \)
97 \( 1 - 2.24iT - 97T^{2} \)
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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

−11.42858734976111189381039273728, −10.69215615457489155075092886034, −9.528136080784193357151040197722, −9.025614276879992661534536723333, −7.890114994299573337157803115613, −7.10875433680665381091824467100, −5.57101046205488799015121828469, −3.69364430471320310713302760047, −2.87855356431618703637531534917, −0.17779941530162031926979124300, 2.38611960591566056572757190053, 3.68551566358648944133981751977, 5.82746414636844987160710794559, 6.99492864091334947758360239943, 7.50640479212057468819701814543, 8.550147206217887043633885583661, 9.728926926505562919066657385995, 10.28533244540065680087018928373, 11.53879908261145904970603923472, 12.46729367445092420720111037060

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