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Properties

Label 2-120-120.77-c1-0-18
Degree $2$
Conductor $120$
Sign $0.198 + 0.980i$
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  + (1.11 − 0.864i)2-s + (0.170 − 1.72i)3-s + (0.506 − 1.93i)4-s + (−1.36 + 1.77i)5-s + (−1.29 − 2.07i)6-s + (2.06 + 2.06i)7-s + (−1.10 − 2.60i)8-s + (−2.94 − 0.586i)9-s + (0.00136 + 3.16i)10-s − 0.510·11-s + (−3.24 − 1.20i)12-s + (0.750 + 0.750i)13-s + (4.10 + 0.528i)14-s + (2.81 + 2.65i)15-s + (−3.48 − 1.95i)16-s + (−3.14 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.791 − 0.611i)2-s + (0.0982 − 0.995i)3-s + (0.253 − 0.967i)4-s + (−0.610 + 0.791i)5-s + (−0.530 − 0.847i)6-s + (0.782 + 0.782i)7-s + (−0.390 − 0.920i)8-s + (−0.980 − 0.195i)9-s + (0.000431 + 0.999i)10-s − 0.153·11-s + (−0.937 − 0.346i)12-s + (0.208 + 0.208i)13-s + (1.09 + 0.141i)14-s + (0.727 + 0.685i)15-s + (−0.871 − 0.489i)16-s + (−0.763 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16267 - 0.950837i\)
\(L(\frac12)\) \(\approx\) \(1.16267 - 0.950837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.864i)T \)
3 \( 1 + (-0.170 + 1.72i)T \)
5 \( 1 + (1.36 - 1.77i)T \)
good7 \( 1 + (-2.06 - 2.06i)T + 7iT^{2} \)
11 \( 1 + 0.510T + 11T^{2} \)
13 \( 1 + (-0.750 - 0.750i)T + 13iT^{2} \)
17 \( 1 + (3.14 - 3.14i)T - 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (-2.54 - 2.54i)T + 23iT^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (6.76 - 6.76i)T - 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + (5.95 + 5.95i)T + 43iT^{2} \)
47 \( 1 + (-3.33 + 3.33i)T - 47iT^{2} \)
53 \( 1 + (5.75 - 5.75i)T - 53iT^{2} \)
59 \( 1 + 1.16iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (-7.98 + 7.98i)T - 67iT^{2} \)
71 \( 1 + 5.09iT - 71T^{2} \)
73 \( 1 + (3.20 - 3.20i)T - 73iT^{2} \)
79 \( 1 + 7.31iT - 79T^{2} \)
83 \( 1 + (-4.77 + 4.77i)T - 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
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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

−13.29070233334773036911228539764, −11.97705926480159043151420897692, −11.63539300044132453886187739008, −10.62297155046823761382486323904, −8.925524905768040347093902966566, −7.62532109267150214532807456407, −6.48813289476535918673099142238, −5.26400442406829410998135509842, −3.42350360609839962217473166867, −2.01028538203947900318104934387, 3.43550736402308166291778606064, 4.63125321007895621233262274311, 5.27721991753884187374161880759, 7.22518744718147832966413744948, 8.238332449873512320083811218377, 9.230785472489144334960243745858, 10.93376510061583243033972794924, 11.58217855680395537978459905027, 12.86725966878911565946392490950, 13.94294859174323265625263801187

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