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Properties

Label 2-120-40.29-c1-0-11
Degree $2$
Conductor $120$
Sign $-0.850 + 0.525i$
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  + (0.321 − 1.37i)2-s − 3-s + (−1.79 − 0.884i)4-s + (−2.11 − 0.726i)5-s + (−0.321 + 1.37i)6-s − 4.05i·7-s + (−1.79 + 2.18i)8-s + 9-s + (−1.67 + 2.67i)10-s − 0.985i·11-s + (1.79 + 0.884i)12-s + 4.94·13-s + (−5.58 − 1.30i)14-s + (2.11 + 0.726i)15-s + (2.43 + 3.17i)16-s + 4.52i·17-s + ⋯
L(s)  = 1  + (0.227 − 0.973i)2-s − 0.577·3-s + (−0.896 − 0.442i)4-s + (−0.945 − 0.324i)5-s + (−0.131 + 0.562i)6-s − 1.53i·7-s + (−0.634 + 0.773i)8-s + 0.333·9-s + (−0.530 + 0.847i)10-s − 0.297i·11-s + (0.517 + 0.255i)12-s + 1.37·13-s + (−1.49 − 0.348i)14-s + (0.546 + 0.187i)15-s + (0.608 + 0.793i)16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.850 + 0.525i$
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: \((2,\ 120,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202532 - 0.713608i\)
\(L(\frac12)\) \(\approx\) \(0.202532 - 0.713608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.321 + 1.37i)T \)
3 \( 1 + T \)
5 \( 1 + (2.11 + 0.726i)T \)
good7 \( 1 + 4.05iT - 7T^{2} \)
11 \( 1 + 0.985iT - 11T^{2} \)
13 \( 1 - 4.94T + 13T^{2} \)
17 \( 1 - 4.52iT - 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 + 3.53iT - 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + 0.945T + 37T^{2} \)
41 \( 1 - 0.568T + 41T^{2} \)
43 \( 1 - 8.45T + 43T^{2} \)
47 \( 1 - 2.60iT - 47T^{2} \)
53 \( 1 - 0.229T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 - 11.0iT - 61T^{2} \)
67 \( 1 + 8.45T + 67T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 + 11.9iT - 73T^{2} \)
79 \( 1 - 3.28T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−13.02783342577903683325405448930, −11.88926788042961248106131378379, −10.88693539315575781013065101364, −10.54044454415802118388705536199, −8.899688926829588927325255856610, −7.73375757972209273827066007344, −6.14296126560268722196977971321, −4.41879866932439474026769339950, −3.74240884645171283901061950817, −0.884039741002023211256200436812, 3.48817147279504785598946044861, 5.06683614032494353272679440030, 6.07284318511331767459595709353, 7.23918524644620620596748114809, 8.414723893450349498911240007613, 9.342041397374065459972169374923, 11.06630298305492221200960211112, 12.02583310401137139492806476256, 12.75196191853688931901251400382, 14.13925826244260216745486578973

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