Properties

Label 2-120-120.53-c1-0-10
Degree $2$
Conductor $120$
Sign $0.872 + 0.488i$
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  + (−1.39 − 0.250i)2-s + (1.40 − 1.01i)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 + (−2.43 − 1.43i)8-s + (0.925 − 2.85i)9-s + (−3.07 − 0.720i)10-s + 2.28·11-s + (3.33 − 0.934i)12-s + (1.05 − 1.05i)13-s + (3.76 − 2.61i)14-s + (3.24 − 2.11i)15-s + (3.03 + 2.61i)16-s + (−3.04 − 3.04i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.176i)2-s + (0.808 − 0.588i)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 + (−0.861 − 0.508i)8-s + (0.308 − 0.951i)9-s + (−0.973 − 0.227i)10-s + 0.688·11-s + (0.962 − 0.269i)12-s + (0.292 − 0.292i)13-s + (1.00 − 0.698i)14-s + (0.838 − 0.545i)15-s + (0.757 + 0.652i)16-s + (−0.738 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.872 + 0.488i$
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: \((2,\ 120,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932013 - 0.243015i\)
\(L(\frac12)\) \(\approx\) \(0.932013 - 0.243015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.250i)T \)
3 \( 1 + (-1.40 + 1.01i)T \)
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

−13.17146877159375124663973737056, −12.50939363739621681444039935829, −11.25780014967893480023334707874, −9.686453319922357650124904009553, −9.307372894180698234534893832342, −8.311448092807140954591634973140, −6.81557195336565740962971736110, −6.09571454096829543607414925934, −3.16245939904202339600401736869, −1.92527567049929322060848444190, 2.12765127531634997350686250376, 3.94831551909607731588627011211, 6.07242452460116622549392451520, 7.07164240423084019447456480909, 8.591133198086103746534217078413, 9.302073308290067414280269068661, 10.22780555559437265431701252379, 10.83790411374374776967546709305, 12.70136277383995675400086586673, 13.78414037237018989453622980029

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