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Properties

Label 2-120-120.53-c1-0-19
Degree $2$
Conductor $120$
Sign $-0.437 + 0.899i$
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 + (−2.22 + 0.224i)5-s − 2.44·6-s + (1.44 − 1.44i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−2 + 2.44i)10-s + 6.44·11-s + (−2.44 + 2.44i)12-s − 2.89i·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 + (−0.994 + 0.100i)5-s − 0.999·6-s + (0.547 − 0.547i)7-s + (−0.707 − 0.707i)8-s + 0.999i·9-s + (−0.632 + 0.774i)10-s + 1.94·11-s + (−0.707 + 0.707i)12-s − 0.774i·14-s + (0.774 + 0.632i)15-s − 16-s + (0.707 + 0.707i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(0.958204\)
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: \((2,\ 120,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598916 - 0.957518i\)
\(L(\frac12)\) \(\approx\) \(0.598916 - 0.957518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.224i)T \)
good7 \( 1 + (-1.44 + 1.44i)T - 7iT^{2} \)
11 \( 1 - 6.44T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 9.34iT - 29T^{2} \)
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 \( 1 + 0.651iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-2.10 - 2.10i)T + 73iT^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + (4 + 4i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (10.7 - 10.7i)T - 97iT^{2} \)
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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

−12.85571072580321806214291282299, −12.00646262477686618990025325901, −11.37795685467332643279671168680, −10.62693881036321143214510995094, −8.947596864384573830521560302011, −7.34089307189216766017915737471, −6.43167306460902167475225756258, −4.85714829861384773060946604711, −3.71876007951099834816013755585, −1.33236766640431911624838688485, 3.72901146605040982718353943067, 4.54417777661049656713471694880, 5.87515932874913272184166221762, 6.97958241767360526678872569909, 8.397913646367035705187934380753, 9.364097183916475805325918138898, 11.30352520736265410036947355331, 11.73102164647872655524427352244, 12.52318574756639498320096223145, 14.18780064708692960438263801847

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