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Properties

Label 2-120-5.2-c2-0-3
Degree $2$
Conductor $120$
Sign $0.793 + 0.608i$
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 + (4.67 − 1.77i)5-s + (−0.550 − 0.550i)7-s − 2.99i·9-s − 1.55·11-s + (9.55 − 9.55i)13-s + (3.55 − 7.89i)15-s + (11.1 + 11.1i)17-s − 12.6i·19-s − 1.34·21-s + (−21.3 + 21.3i)23-s + (18.6 − 16.5i)25-s + (−3.67 − 3.67i)27-s + 44.0i·29-s − 44.4·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.934 − 0.355i)5-s + (−0.0786 − 0.0786i)7-s − 0.333i·9-s − 0.140·11-s + (0.734 − 0.734i)13-s + (0.236 − 0.526i)15-s + (0.655 + 0.655i)17-s − 0.668i·19-s − 0.0642·21-s + (−0.928 + 0.928i)23-s + (0.747 − 0.663i)25-s + (−0.136 − 0.136i)27-s + 1.51i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.793 + 0.608i$
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: \((2,\ 120,\ (\ :1),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.66537 - 0.565091i\)
\(L(\frac12)\) \(\approx\) \(1.66537 - 0.565091i\)
\(L(2)\) 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 \( 1 + (-4.67 + 1.77i)T \)
good7 \( 1 + (0.550 + 0.550i)T + 49iT^{2} \)
11 \( 1 + 1.55T + 121T^{2} \)
13 \( 1 + (-9.55 + 9.55i)T - 169iT^{2} \)
17 \( 1 + (-11.1 - 11.1i)T + 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
23 \( 1 + (21.3 - 21.3i)T - 529iT^{2} \)
29 \( 1 - 44.0iT - 841T^{2} \)
31 \( 1 + 44.4T + 961T^{2} \)
37 \( 1 + (-20.6 - 20.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2T + 1.68e3T^{2} \)
43 \( 1 + (36.2 - 36.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-42.5 - 42.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (54.4 - 54.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 59.8T + 3.72e3T^{2} \)
67 \( 1 + (-81.2 - 81.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 87.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.9 + 75.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (-41.0 + 41.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 52.2iT - 7.92e3T^{2} \)
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

−13.16183776100871710905008415863, −12.44801318053306909881499480973, −10.95352079939728598475522034957, −9.881028129300045554492653294753, −8.853552437916680110674959254323, −7.79609326397277276349129345744, −6.37367389572388900220400089167, −5.29026472885458282223952479571, −3.33940829845646052058592555430, −1.55401693702323615209286766219, 2.15423793253732133609942230130, 3.78149431550782758762114396197, 5.44572819254134851247411342337, 6.59348269299264904442030785840, 8.072299062861534687226918380689, 9.297358681094068523176602676169, 10.05332181485976553886961112475, 11.09629144816035030029404128650, 12.38668864247846701859861151304, 13.69041249360254073281644350576

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