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Properties

Label 2-120-120.53-c1-0-15
Degree $2$
Conductor $120$
Sign $0.793 + 0.608i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
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 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (0.224 − 2.22i)5-s + 2.44·6-s + (−3.44 + 3.44i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−2 − 2.44i)10-s + 1.55·11-s + (2.44 − 2.44i)12-s + 6.89i·14-s + (2.99 − 2.44i)15-s − 4·16-s + (2.99 + 2.99i)18-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (0.100 − 0.994i)5-s + 0.999·6-s + (−1.30 + 1.30i)7-s + (−0.707 − 0.707i)8-s + 0.999i·9-s + (−0.632 − 0.774i)10-s + 0.467·11-s + (0.707 − 0.707i)12-s + 1.84i·14-s + (0.774 − 0.632i)15-s − 16-s + (0.707 + 0.707i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52737 - 0.518265i\)
\(L(\frac12)\) \(\approx\) \(1.52737 - 0.518265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-0.224 + 2.22i)T \)
good7 \( 1 + (3.44 - 3.44i)T - 7iT^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 5.34iT - 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 + 15.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11.8 - 11.8i)T + 73iT^{2} \)
79 \( 1 - 14.6iT - 79T^{2} \)
83 \( 1 + (4 + 4i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-8.79 + 8.79i)T - 97iT^{2} \)
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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

−13.22533647640015539979488912958, −12.50699243450377762331485054253, −11.56085103190057420058583468490, −9.935841918997355946545608675042, −9.416200152755732205001827323502, −8.521603896613374908086709270396, −6.22322758073195348110990522134, −5.11850981422173399394926350098, −3.78877405540473134226226879488, −2.44366999295522310192634292701, 2.99776426019454343936143168879, 3.90086673055437579412211919535, 6.28034123102157793179296107663, 6.88596866754836332506236089286, 7.67495529477065139403961646919, 9.187057553234918187388590182671, 10.41910752839780795507238379405, 11.91065649893828646606668127295, 13.02512040437146557062300299201, 13.69102926914220212314699249863

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