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Properties

Label 2-120-40.29-c1-0-4
Degree $2$
Conductor $120$
Sign $0.331 + 0.943i$
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  + (−1.16 − 0.806i)2-s − 3-s + (0.699 + 1.87i)4-s + (1.86 − 1.23i)5-s + (1.16 + 0.806i)6-s − 0.746i·7-s + (0.699 − 2.74i)8-s + 9-s + (−3.16 − 0.0603i)10-s − 5.36i·11-s + (−0.699 − 1.87i)12-s + 2.92·13-s + (−0.601 + 0.866i)14-s + (−1.86 + 1.23i)15-s + (−3.02 + 2.62i)16-s − 2.13i·17-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)2-s − 0.577·3-s + (0.349 + 0.936i)4-s + (0.832 − 0.554i)5-s + (0.474 + 0.329i)6-s − 0.282i·7-s + (0.247 − 0.968i)8-s + 0.333·9-s + (−0.999 − 0.0190i)10-s − 1.61i·11-s + (−0.201 − 0.540i)12-s + 0.811·13-s + (−0.160 + 0.231i)14-s + (−0.480 + 0.320i)15-s + (−0.755 + 0.655i)16-s − 0.517i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577445 - 0.409105i\)
\(L(\frac12)\) \(\approx\) \(0.577445 - 0.409105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.806i)T \)
3 \( 1 + T \)
5 \( 1 + (-1.86 + 1.23i)T \)
good7 \( 1 + 0.746iT - 7T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 7.49iT - 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 - 1.07T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 7.44T + 43T^{2} \)
47 \( 1 + 1.73iT - 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 - 6.85iT - 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 - 7.44T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 - 5.85T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−13.27419523193698749204898973369, −11.97253805208643580075223517558, −11.11267405859808680403547554768, −10.21311305670205304083299185613, −9.121898536469972246729640593130, −8.207947867301455284859672570407, −6.63721449314990582705857205057, −5.42270100349333544655125648144, −3.46030293882061327036993626990, −1.24970307222334753202882376774, 2.03227801394467690408670626568, 4.87299565805114013270565354649, 6.22691155196623033128677063676, 6.86981232137470216174228105715, 8.324658537100357035834056499624, 9.665990322452506507307677099869, 10.28650304374129121761879426825, 11.30631217974284277635640847731, 12.61664424480836437054393732659, 13.86332610378610376834268972115

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