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Properties

Label 2-120-24.11-c3-0-41
Degree $2$
Conductor $120$
Sign $0.459 + 0.888i$
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  + (2.03 − 1.96i)2-s + (5.02 + 1.30i)3-s + (0.292 − 7.99i)4-s + 5·5-s + (12.8 − 7.21i)6-s − 8.16i·7-s + (−15.0 − 16.8i)8-s + (23.6 + 13.1i)9-s + (10.1 − 9.81i)10-s + 16.0i·11-s + (11.8 − 39.8i)12-s − 41.4i·13-s + (−16.0 − 16.6i)14-s + (25.1 + 6.51i)15-s + (−63.8 − 4.67i)16-s + 83.3i·17-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.968 + 0.250i)3-s + (0.0365 − 0.999i)4-s + 0.447·5-s + (0.871 − 0.491i)6-s − 0.440i·7-s + (−0.667 − 0.744i)8-s + (0.874 + 0.485i)9-s + (0.321 − 0.310i)10-s + 0.438i·11-s + (0.286 − 0.958i)12-s − 0.884i·13-s + (−0.305 − 0.317i)14-s + (0.432 + 0.112i)15-s + (−0.997 − 0.0730i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.459 + 0.888i$
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: \((2,\ 120,\ (\ :3/2),\ 0.459 + 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.77581 - 1.69011i\)
\(L(\frac12)\) \(\approx\) \(2.77581 - 1.69011i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.03 + 1.96i)T \)
3 \( 1 + (-5.02 - 1.30i)T \)
5 \( 1 - 5T \)
good7 \( 1 + 8.16iT - 343T^{2} \)
11 \( 1 - 16.0iT - 1.33e3T^{2} \)
13 \( 1 + 41.4iT - 2.19e3T^{2} \)
17 \( 1 - 83.3iT - 4.91e3T^{2} \)
19 \( 1 - 54.9T + 6.85e3T^{2} \)
23 \( 1 + 66.4T + 1.21e4T^{2} \)
29 \( 1 + 153.T + 2.43e4T^{2} \)
31 \( 1 + 11.4iT - 2.97e4T^{2} \)
37 \( 1 - 245. iT - 5.06e4T^{2} \)
41 \( 1 + 14.3iT - 6.89e4T^{2} \)
43 \( 1 + 485.T + 7.95e4T^{2} \)
47 \( 1 - 70.6T + 1.03e5T^{2} \)
53 \( 1 - 514.T + 1.48e5T^{2} \)
59 \( 1 - 488. iT - 2.05e5T^{2} \)
61 \( 1 - 886. iT - 2.26e5T^{2} \)
67 \( 1 + 780.T + 3.00e5T^{2} \)
71 \( 1 + 520.T + 3.57e5T^{2} \)
73 \( 1 - 490.T + 3.89e5T^{2} \)
79 \( 1 + 1.15e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.24e3iT - 7.04e5T^{2} \)
97 \( 1 - 296.T + 9.12e5T^{2} \)
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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

−13.18419220735141604880467679837, −11.97273917200662705674114498096, −10.41442470871784673869949913198, −10.05446699254574916705167576380, −8.751740199221549349493321255086, −7.33539937453298994604256404661, −5.74824010871323510327857588660, −4.33245686562510984743337906148, −3.15115695473592181450886229770, −1.66563782669911959391888135697, 2.31875881977033604554297002870, 3.69067190161948048840272130984, 5.23890535911488812103674266153, 6.59376675787549892876799899686, 7.60064175991947861008860370358, 8.792345089312793320547862443788, 9.569864293250865005769243302458, 11.49188316497648116720247674097, 12.45622736521021980374782630112, 13.61914534864104287820340606451

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