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Properties

Label 2-120-120.53-c1-0-13
Degree $2$
Conductor $120$
Sign $0.972 + 0.234i$
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  + (1.41 + 0.0941i)2-s + (−1.68 − 0.416i)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 + (2.77 + 0.561i)8-s + (2.65 + 1.40i)9-s + (2.43 − 2.02i)10-s − 2.63·11-s + (−3.22 − 1.27i)12-s + (−3.49 + 3.49i)13-s + (−0.544 + 0.476i)14-s + (−3.36 + 1.91i)15-s + (3.85 + 1.05i)16-s + (−3.61 − 3.61i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0665i)2-s + (−0.970 − 0.240i)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 + (0.980 + 0.198i)8-s + (0.884 + 0.466i)9-s + (0.769 − 0.638i)10-s − 0.794·11-s + (−0.930 − 0.367i)12-s + (−0.968 + 0.968i)13-s + (−0.145 + 0.127i)14-s + (−0.869 + 0.494i)15-s + (0.964 + 0.263i)16-s + (−0.876 − 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.972 + 0.234i$
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: \((2,\ 120,\ (\ :1/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47925 - 0.175761i\)
\(L(\frac12)\) \(\approx\) \(1.47925 - 0.175761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0941i)T \)
3 \( 1 + (1.68 + 0.416i)T \)
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

−13.39005039574373496318138410567, −12.41765741514241666190778776085, −11.76829120942313902485500131948, −10.58649482070175993340597522348, −9.445867724157522922402985093476, −7.56438374791888586325921907142, −6.47052091506511208879437123507, −5.35488946864313463135466148459, −4.58305846880449325612744215944, −2.15523670621358233049512183517, 2.56635425807087230973151481604, 4.39229959424836275096459909554, 5.65010560185608275944114146847, 6.41656676754295628925680577809, 7.64158520779358682973407520603, 10.03741051194394526131518942214, 10.45145047676257193748751156965, 11.48475266298251613884225277167, 12.68406320949440557910723493442, 13.26671572515331130111905494450

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