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Properties

Label 2-120-120.77-c1-0-19
Degree $2$
Conductor $120$
Sign $-0.764 + 0.644i$
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  + (0.250 − 1.39i)2-s + (−1.01 − 1.40i)3-s + (−1.87 − 0.696i)4-s + (2.23 − 0.116i)5-s + (−2.20 + 1.06i)6-s + (−2.29 − 2.29i)7-s + (−1.43 + 2.43i)8-s + (−0.925 + 2.85i)9-s + (0.396 − 3.13i)10-s + 2.28·11-s + (0.934 + 3.33i)12-s + (−1.05 − 1.05i)13-s + (−3.76 + 2.61i)14-s + (−2.43 − 3.00i)15-s + (3.03 + 2.61i)16-s + (3.04 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.176 − 0.984i)2-s + (−0.588 − 0.808i)3-s + (−0.937 − 0.348i)4-s + (0.998 − 0.0519i)5-s + (−0.900 + 0.435i)6-s + (−0.865 − 0.865i)7-s + (−0.508 + 0.861i)8-s + (−0.308 + 0.951i)9-s + (0.125 − 0.992i)10-s + 0.688·11-s + (0.269 + 0.962i)12-s + (−0.292 − 0.292i)13-s + (−1.00 + 0.698i)14-s + (−0.629 − 0.777i)15-s + (0.757 + 0.652i)16-s + (0.738 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.764 + 0.644i$
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: \((2,\ 120,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324170 - 0.886687i\)
\(L(\frac12)\) \(\approx\) \(0.324170 - 0.886687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.250 + 1.39i)T \)
3 \( 1 + (1.01 + 1.40i)T \)
5 \( 1 + (-2.23 + 0.116i)T \)
good7 \( 1 + (2.29 + 2.29i)T + 7iT^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (1.05 + 1.05i)T + 13iT^{2} \)
17 \( 1 + (-3.04 + 3.04i)T - 17iT^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
23 \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \)
29 \( 1 + 2.71iT - 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + (2.31 - 2.31i)T - 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (-1.16 - 1.16i)T + 43iT^{2} \)
47 \( 1 + (1.83 - 1.83i)T - 47iT^{2} \)
53 \( 1 + (5.82 - 5.82i)T - 53iT^{2} \)
59 \( 1 - 7.41iT - 59T^{2} \)
61 \( 1 + 8.97iT - 61T^{2} \)
67 \( 1 + (8.66 - 8.66i)T - 67iT^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (-1.83 + 1.83i)T - 73iT^{2} \)
79 \( 1 - 8.28iT - 79T^{2} \)
83 \( 1 + (-5.27 + 5.27i)T - 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (2.79 + 2.79i)T + 97iT^{2} \)
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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

−13.11116443352523575661527797859, −12.13739212049086769910429785757, −11.11366399846570259697157666470, −10.02507606207812451567283327223, −9.309348408774495610188651341718, −7.46352304034550533453790222394, −6.19387156193611286087044171318, −5.05801583064239343517014288521, −3.10269244950074462248904383062, −1.22241965265579175325311963162, 3.46944298377312440836996918197, 5.16945843054804560102665793857, 5.95393182243141286790749156923, 6.86792575835271986540020047802, 8.949391690741773214327471578705, 9.388231513153627403344184152988, 10.44933274413554656061084476445, 12.13191435572605716896383005152, 12.84467456322552778430566841996, 14.25450614861670820298581599549

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