Properties

Label 2-120-120.53-c1-0-18
Degree $2$
Conductor $120$
Sign $-0.796 + 0.604i$
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  + (−0.0941 − 1.41i)2-s + (−0.416 − 1.68i)3-s + (−1.98 + 0.265i)4-s + (1.62 − 1.54i)5-s + (−2.33 + 0.746i)6-s + (−0.361 + 0.361i)7-s + (0.561 + 2.77i)8-s + (−2.65 + 1.40i)9-s + (−2.32 − 2.14i)10-s − 2.63·11-s + (1.27 + 3.22i)12-s + (3.49 − 3.49i)13-s + (0.544 + 0.476i)14-s + (−3.26 − 2.08i)15-s + (3.85 − 1.05i)16-s + (3.61 + 3.61i)17-s + ⋯
L(s)  = 1  + (−0.0665 − 0.997i)2-s + (−0.240 − 0.970i)3-s + (−0.991 + 0.132i)4-s + (0.724 − 0.688i)5-s + (−0.952 + 0.304i)6-s + (−0.136 + 0.136i)7-s + (0.198 + 0.980i)8-s + (−0.884 + 0.466i)9-s + (−0.735 − 0.677i)10-s − 0.794·11-s + (0.367 + 0.930i)12-s + (0.968 − 0.968i)13-s + (0.145 + 0.127i)14-s + (−0.842 − 0.538i)15-s + (0.964 − 0.263i)16-s + (0.876 + 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.796 + 0.604i$
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: \((2,\ 120,\ (\ :1/2),\ -0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294231 - 0.874608i\)
\(L(\frac12)\) \(\approx\) \(0.294231 - 0.874608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0941 + 1.41i)T \)
3 \( 1 + (0.416 + 1.68i)T \)
5 \( 1 + (-1.62 + 1.54i)T \)
good7 \( 1 + (0.361 - 0.361i)T - 7iT^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + (-3.49 + 3.49i)T - 13iT^{2} \)
17 \( 1 + (-3.61 - 3.61i)T + 17iT^{2} \)
19 \( 1 + 0.672T + 19T^{2} \)
23 \( 1 + (-4.31 + 4.31i)T - 23iT^{2} \)
29 \( 1 - 4.76iT - 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 - 4.10iT - 41T^{2} \)
43 \( 1 + (7.57 - 7.57i)T - 43iT^{2} \)
47 \( 1 + (-0.987 - 0.987i)T + 47iT^{2} \)
53 \( 1 + (0.646 + 0.646i)T + 53iT^{2} \)
59 \( 1 + 4.92iT - 59T^{2} \)
61 \( 1 - 6.07iT - 61T^{2} \)
67 \( 1 + (0.349 + 0.349i)T + 67iT^{2} \)
71 \( 1 - 8.63iT - 71T^{2} \)
73 \( 1 + (11.3 + 11.3i)T + 73iT^{2} \)
79 \( 1 - 4.07iT - 79T^{2} \)
83 \( 1 + (-8.53 - 8.53i)T + 83iT^{2} \)
89 \( 1 + 6.58T + 89T^{2} \)
97 \( 1 + (0.660 - 0.660i)T - 97iT^{2} \)
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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

−12.87337239476214164909556065430, −12.39946957281253113651886712658, −10.98426605801230822035672887092, −10.18746332368654384452540796957, −8.724018154922857901939226054798, −8.032375744686342222596686895644, −6.07525383920787500569993230532, −5.08717132526540000131402852927, −2.91984569372282031261810781133, −1.25486116545243373921808275420, 3.45276236132942520383021396879, 5.03717212265132506866764312114, 6.01259595739460948102861256608, 7.14715141469026893008876660705, 8.679519008667815073186260562454, 9.685707952967844404653724258956, 10.41063916130411454167731520132, 11.63524942833902705227138812523, 13.45383738973057389083129602606, 13.96209056768667841840436490781

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