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Properties

Label 2-120-5.2-c2-0-4
Degree $2$
Conductor $120$
Sign $-0.437 + 0.899i$
Analytic cond. $3.26976$
Root an. cond. $1.80824$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + (−2.67 − 4.22i)5-s + (−5.44 − 5.44i)7-s − 2.99i·9-s − 6.44·11-s + (14.4 − 14.4i)13-s + (8.44 + 1.89i)15-s + (−23.1 − 23.1i)17-s + 16.6i·19-s + 13.3·21-s + (−6.65 + 6.65i)23-s + (−10.6 + 22.5i)25-s + (3.67 + 3.67i)27-s − 0.0454i·29-s + 4.49·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.534 − 0.844i)5-s + (−0.778 − 0.778i)7-s − 0.333i·9-s − 0.586·11-s + (1.11 − 1.11i)13-s + (0.563 + 0.126i)15-s + (−1.36 − 1.36i)17-s + 0.878i·19-s + 0.635·21-s + (−0.289 + 0.289i)23-s + (−0.427 + 0.903i)25-s + (0.136 + 0.136i)27-s − 0.00156i·29-s + 0.144·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(3.26976\)
Root analytic conductor: \(1.80824\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.349648 - 0.559000i\)
\(L(\frac12)\) \(\approx\) \(0.349648 - 0.559000i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (2.67 + 4.22i)T \)
good7 \( 1 + (5.44 + 5.44i)T + 49iT^{2} \)
11 \( 1 + 6.44T + 121T^{2} \)
13 \( 1 + (-14.4 + 14.4i)T - 169iT^{2} \)
17 \( 1 + (23.1 + 23.1i)T + 289iT^{2} \)
19 \( 1 - 16.6iT - 361T^{2} \)
23 \( 1 + (6.65 - 6.65i)T - 529iT^{2} \)
29 \( 1 + 0.0454iT - 841T^{2} \)
31 \( 1 - 4.49T + 961T^{2} \)
37 \( 1 + (-35.3 - 35.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 20.2T + 1.68e3T^{2} \)
43 \( 1 + (-32.2 + 32.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (50.5 + 50.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (5.50 - 5.50i)T - 2.80e3iT^{2} \)
59 \( 1 - 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (85.2 + 85.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 48.4T + 5.04e3T^{2} \)
73 \( 1 + (21.9 - 21.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-94.9 + 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (37 + 37i)T + 9.40e3iT^{2} \)
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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

−13.06280210856661600064619771301, −11.82636397098649655125865906719, −10.80987146066242348552000631033, −9.842843020306348632128424359675, −8.650527401604275245037241719039, −7.47651951525119027376790684495, −6.02309218788136874746594964545, −4.71178996957702727443482843903, −3.47483302731253806820562059774, −0.47699208389751010586953856536, 2.46574473788976158320914683497, 4.14467968776531820176987018549, 6.12126431601138369023957258013, 6.67897667791939246832837192129, 8.144805644107246322127718800929, 9.282236597770459583675542149307, 10.83318037196763272646149501808, 11.32879676621895578305668620158, 12.60295957199261079407143049443, 13.34969371312222081175613772606

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