Properties

Label 2-120-8.5-c1-0-2
Degree $2$
Conductor $120$
Sign $0.996 - 0.0864i$
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  + (−0.671 + 1.24i)2-s i·3-s + (−1.09 − 1.67i)4-s i·5-s + (1.24 + 0.671i)6-s + 4.68·7-s + (2.81 − 0.244i)8-s − 9-s + (1.24 + 0.671i)10-s + 2.29i·11-s + (−1.67 + 1.09i)12-s − 4.97i·13-s + (−3.14 + 5.83i)14-s − 15-s + (−1.58 + 3.67i)16-s − 2.97·17-s + ⋯
L(s)  = 1  + (−0.474 + 0.880i)2-s − 0.577i·3-s + (−0.549 − 0.835i)4-s − 0.447i·5-s + (0.508 + 0.274i)6-s + 1.77·7-s + (0.996 − 0.0864i)8-s − 0.333·9-s + (0.393 + 0.212i)10-s + 0.691i·11-s + (−0.482 + 0.317i)12-s − 1.38i·13-s + (−0.840 + 1.55i)14-s − 0.258·15-s + (−0.396 + 0.917i)16-s − 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.996 - 0.0864i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.996 - 0.0864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908016 + 0.0393430i\)
\(L(\frac12)\) \(\approx\) \(0.908016 + 0.0393430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.671 - 1.24i)T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
good7 \( 1 - 4.68T + 7T^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 + 4.97iT - 13T^{2} \)
17 \( 1 + 2.97T + 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 - 4.39iT - 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 9.37iT - 43T^{2} \)
47 \( 1 - 7.27T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 - 4.58iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 3.95T + 97T^{2} \)
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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

−13.64242040333973312688654894574, −12.65542876876037515388157823281, −11.33052370520734268777314314752, −10.29455132753295714519383184835, −8.780399302700909981601556615236, −8.041675282632607199380252503055, −7.21449140090446490598010874412, −5.59585796279181435083924183329, −4.70243546252100533273111736366, −1.55441951206953152948658851411, 2.07685081005802612600182856735, 3.93951373413611225449352306003, 5.05224771178053274801949782614, 7.17919514397083014867525424051, 8.505218967351777115895583711927, 9.182603588282188860778720184797, 10.72029639061962227568002912757, 11.17966274914024197691899749621, 11.91175547419068285088247608161, 13.61889321833959803791498568261

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