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Properties

Label 2-120-120.77-c3-0-2
Degree $2$
Conductor $120$
Sign $-0.968 + 0.248i$
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  + (−2 − 2i)2-s + (−3.67 + 3.67i)3-s + 8i·4-s + (−3.32 + 10.6i)5-s + 14.6·6-s + (9.65 + 9.65i)7-s + (16 − 16i)8-s − 27i·9-s + (28 − 14.6i)10-s − 47.4·11-s + (−29.3 − 29.3i)12-s − 38.6i·14-s + (−26.9 − 51.4i)15-s − 64·16-s + (−54 + 54i)18-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.297 + 0.954i)5-s + 0.999·6-s + (0.521 + 0.521i)7-s + (0.707 − 0.707i)8-s − i·9-s + (0.885 − 0.464i)10-s − 1.30·11-s + (−0.707 − 0.707i)12-s − 0.736i·14-s + (−0.464 − 0.885i)15-s − 16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.968 + 0.248i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0130544 - 0.103249i\)
\(L(\frac12)\) \(\approx\) \(0.0130544 - 0.103249i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
3 \( 1 + (3.67 - 3.67i)T \)
5 \( 1 + (3.32 - 10.6i)T \)
good7 \( 1 + (-9.65 - 9.65i)T + 343iT^{2} \)
11 \( 1 + 47.4T + 1.33e3T^{2} \)
13 \( 1 + 2.19e3iT^{2} \)
17 \( 1 - 4.91e3iT^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 1.21e4iT^{2} \)
29 \( 1 + 312. iT - 2.43e4T^{2} \)
31 \( 1 + 338.T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4iT^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 7.95e4iT^{2} \)
47 \( 1 - 1.03e5iT^{2} \)
53 \( 1 + (-360. + 360. i)T - 1.48e5iT^{2} \)
59 \( 1 - 899. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5iT^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (763. - 763. i)T - 3.89e5iT^{2} \)
79 \( 1 - 308. iT - 4.93e5T^{2} \)
83 \( 1 + (868 - 868i)T - 5.71e5iT^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + (-1.19e3 - 1.19e3i)T + 9.12e5iT^{2} \)
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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

−13.27015952761502944922078242428, −12.02205241839574485141460700410, −11.27396639348325094990504497311, −10.54873204946661654963510827758, −9.705719910608478520077198908861, −8.339898851765800389980323611132, −7.20607596472202746268658492992, −5.58504697290502058351313414914, −4.00212190583182385064155963610, −2.52883773183139141570356207795, 0.07587764675362501251942622142, 1.54433763187092646403498301697, 4.84605251233577030099586435630, 5.60237259640998413285340567490, 7.21011272407411837351803454659, 7.86301827116210954863857899850, 8.922619634932462451964436626998, 10.46903385830415368907839869549, 11.17485898637323665034612810432, 12.55522510246796760262041183975

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