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Properties

Label 2-120-24.11-c3-0-34
Degree $2$
Conductor $120$
Sign $-0.640 + 0.767i$
Analytic cond. $7.08022$
Root an. cond. $2.66087$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.54 − 2.37i)2-s + (3.91 − 3.41i)3-s + (−3.25 + 7.30i)4-s + 5·5-s + (−14.1 − 4.01i)6-s − 2.12i·7-s + (22.3 − 3.54i)8-s + (3.61 − 26.7i)9-s + (−7.70 − 11.8i)10-s − 49.3i·11-s + (12.2 + 39.7i)12-s − 20.7i·13-s + (−5.03 + 3.26i)14-s + (19.5 − 17.0i)15-s + (−42.8 − 47.5i)16-s + 22.7i·17-s + ⋯
L(s)  = 1  + (−0.544 − 0.838i)2-s + (0.752 − 0.658i)3-s + (−0.406 + 0.913i)4-s + 0.447·5-s + (−0.962 − 0.273i)6-s − 0.114i·7-s + (0.987 − 0.156i)8-s + (0.133 − 0.990i)9-s + (−0.243 − 0.375i)10-s − 1.35i·11-s + (0.295 + 0.955i)12-s − 0.443i·13-s + (−0.0960 + 0.0623i)14-s + (0.336 − 0.294i)15-s + (−0.669 − 0.742i)16-s + 0.324i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.640 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.646706 - 1.38162i\)
\(L(\frac12)\) \(\approx\) \(0.646706 - 1.38162i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.54 + 2.37i)T \)
3 \( 1 + (-3.91 + 3.41i)T \)
5 \( 1 - 5T \)
good7 \( 1 + 2.12iT - 343T^{2} \)
11 \( 1 + 49.3iT - 1.33e3T^{2} \)
13 \( 1 + 20.7iT - 2.19e3T^{2} \)
17 \( 1 - 22.7iT - 4.91e3T^{2} \)
19 \( 1 - 63.4T + 6.85e3T^{2} \)
23 \( 1 + 147.T + 1.21e4T^{2} \)
29 \( 1 + 3.23T + 2.43e4T^{2} \)
31 \( 1 + 133. iT - 2.97e4T^{2} \)
37 \( 1 - 257. iT - 5.06e4T^{2} \)
41 \( 1 + 73.9iT - 6.89e4T^{2} \)
43 \( 1 - 378.T + 7.95e4T^{2} \)
47 \( 1 - 469.T + 1.03e5T^{2} \)
53 \( 1 + 602.T + 1.48e5T^{2} \)
59 \( 1 - 637. iT - 2.05e5T^{2} \)
61 \( 1 - 336. iT - 2.26e5T^{2} \)
67 \( 1 - 684.T + 3.00e5T^{2} \)
71 \( 1 - 898.T + 3.57e5T^{2} \)
73 \( 1 + 335.T + 3.89e5T^{2} \)
79 \( 1 + 814. iT - 4.93e5T^{2} \)
83 \( 1 - 371. iT - 5.71e5T^{2} \)
89 \( 1 - 223. iT - 7.04e5T^{2} \)
97 \( 1 + 1.63e3T + 9.12e5T^{2} \)
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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

−12.61466533069724710940884206537, −11.64012702963760232337727032001, −10.45026991223168347571033421626, −9.388001473114058603018936956825, −8.428687331363191371790516649543, −7.56968892648617439767261942985, −5.97501047919283423554840037658, −3.76247199173800330186081955297, −2.54301599558947515882563626523, −0.945133859746665923050653734096, 2.05031824637394733326060585394, 4.27688063984381039569103708268, 5.45448227837974962774097151538, 7.01796743386860071173271952614, 8.007618503657859883679865090820, 9.313073822356010145791807280152, 9.729433081712425793384217804286, 10.82358056165969119196230153339, 12.53316845263412694587837593583, 14.01223338144689994021489198116

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