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Properties

Label 2-120-120.53-c3-0-37
Degree $2$
Conductor $120$
Sign $0.846 - 0.531i$
Analytic cond. $7.08022$
Root an. cond. $2.66087$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 2.49i)2-s + (5.19 + 0.0934i)3-s + (−4.47 − 6.63i)4-s + (5.77 − 9.57i)5-s + (−7.13 + 12.8i)6-s + (−15.4 + 15.4i)7-s + (22.5 − 2.34i)8-s + (26.9 + 0.970i)9-s + (16.2 + 27.1i)10-s + 24.0·11-s + (−22.6 − 34.8i)12-s + (64.6 − 64.6i)13-s + (−18.1 − 59.2i)14-s + (30.8 − 49.1i)15-s + (−24.0 + 59.3i)16-s + (69.5 + 69.5i)17-s + ⋯
L(s)  = 1  + (−0.469 + 0.882i)2-s + (0.999 + 0.0179i)3-s + (−0.558 − 0.829i)4-s + (0.516 − 0.856i)5-s + (−0.485 + 0.874i)6-s + (−0.836 + 0.836i)7-s + (0.994 − 0.103i)8-s + (0.999 + 0.0359i)9-s + (0.513 + 0.858i)10-s + 0.660·11-s + (−0.543 − 0.839i)12-s + (1.37 − 1.37i)13-s + (−0.345 − 1.13i)14-s + (0.531 − 0.846i)15-s + (−0.375 + 0.926i)16-s + (0.991 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.846 - 0.531i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.846 - 0.531i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.78004 + 0.512345i\)
\(L(\frac12)\) \(\approx\) \(1.78004 + 0.512345i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 2.49i)T \)
3 \( 1 + (-5.19 - 0.0934i)T \)
5 \( 1 + (-5.77 + 9.57i)T \)
good7 \( 1 + (15.4 - 15.4i)T - 343iT^{2} \)
11 \( 1 - 24.0T + 1.33e3T^{2} \)
13 \( 1 + (-64.6 + 64.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (-69.5 - 69.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 0.230T + 6.85e3T^{2} \)
23 \( 1 + (-23.1 + 23.1i)T - 1.21e4iT^{2} \)
29 \( 1 - 110. iT - 2.43e4T^{2} \)
31 \( 1 + 210.T + 2.97e4T^{2} \)
37 \( 1 + (16.5 + 16.5i)T + 5.06e4iT^{2} \)
41 \( 1 - 28.8iT - 6.89e4T^{2} \)
43 \( 1 + (66.7 - 66.7i)T - 7.95e4iT^{2} \)
47 \( 1 + (171. + 171. i)T + 1.03e5iT^{2} \)
53 \( 1 + (375. + 375. i)T + 1.48e5iT^{2} \)
59 \( 1 - 713. iT - 2.05e5T^{2} \)
61 \( 1 + 699. iT - 2.26e5T^{2} \)
67 \( 1 + (401. + 401. i)T + 3.00e5iT^{2} \)
71 \( 1 - 534. iT - 3.57e5T^{2} \)
73 \( 1 + (-124. - 124. i)T + 3.89e5iT^{2} \)
79 \( 1 - 647. iT - 4.93e5T^{2} \)
83 \( 1 + (29.7 + 29.7i)T + 5.71e5iT^{2} \)
89 \( 1 - 250.T + 7.04e5T^{2} \)
97 \( 1 + (209. - 209. i)T - 9.12e5iT^{2} \)
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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

−13.09810649712082960848349176131, −12.68105981549944077135313444300, −10.46389818392228303833414540701, −9.485647306598721603315286366079, −8.742694214577614337666912690531, −8.035684600405136525068003447208, −6.38057800842710973288640210663, −5.44189894502938340721738486554, −3.57860611523453857525371393340, −1.36974381843964581479489286120, 1.53397992778517417041409005165, 3.16445182650125888768670976768, 3.97084364598974993214213317277, 6.62097711747555478212290425329, 7.54462387010996421909749004572, 9.105507204101356097998441958189, 9.623473499102312497415690517930, 10.63632660086879022659837423002, 11.70084542363042761162617829855, 13.15206351978781497719767361288

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