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Properties

Label 2-120-15.2-c3-0-14
Degree $2$
Conductor $120$
Sign $-0.514 + 0.857i$
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  + (1.89 + 4.83i)3-s + (−8.68 − 7.04i)5-s + (−20.0 − 20.0i)7-s + (−19.8 + 18.3i)9-s − 46.6i·11-s + (−20.2 + 20.2i)13-s + (17.6 − 55.3i)15-s + (59.4 − 59.4i)17-s + 81.9i·19-s + (58.8 − 134. i)21-s + (−98.2 − 98.2i)23-s + (25.8 + 122. i)25-s + (−126. − 61.1i)27-s − 18.6·29-s − 278.·31-s + ⋯
L(s)  = 1  + (0.364 + 0.931i)3-s + (−0.776 − 0.629i)5-s + (−1.07 − 1.07i)7-s + (−0.733 + 0.679i)9-s − 1.27i·11-s + (−0.433 + 0.433i)13-s + (0.303 − 0.952i)15-s + (0.848 − 0.848i)17-s + 0.989i·19-s + (0.611 − 1.39i)21-s + (−0.890 − 0.890i)23-s + (0.206 + 0.978i)25-s + (−0.900 − 0.435i)27-s − 0.119·29-s − 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.279148 - 0.492711i\)
\(L(\frac12)\) \(\approx\) \(0.279148 - 0.492711i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.89 - 4.83i)T \)
5 \( 1 + (8.68 + 7.04i)T \)
good7 \( 1 + (20.0 + 20.0i)T + 343iT^{2} \)
11 \( 1 + 46.6iT - 1.33e3T^{2} \)
13 \( 1 + (20.2 - 20.2i)T - 2.19e3iT^{2} \)
17 \( 1 + (-59.4 + 59.4i)T - 4.91e3iT^{2} \)
19 \( 1 - 81.9iT - 6.85e3T^{2} \)
23 \( 1 + (98.2 + 98.2i)T + 1.21e4iT^{2} \)
29 \( 1 + 18.6T + 2.43e4T^{2} \)
31 \( 1 + 278.T + 2.97e4T^{2} \)
37 \( 1 + (81.9 + 81.9i)T + 5.06e4iT^{2} \)
41 \( 1 - 211. iT - 6.89e4T^{2} \)
43 \( 1 + (-168. + 168. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-24.9 + 24.9i)T - 1.03e5iT^{2} \)
53 \( 1 + (-54.7 - 54.7i)T + 1.48e5iT^{2} \)
59 \( 1 - 158.T + 2.05e5T^{2} \)
61 \( 1 - 892.T + 2.26e5T^{2} \)
67 \( 1 + (407. + 407. i)T + 3.00e5iT^{2} \)
71 \( 1 + 286. iT - 3.57e5T^{2} \)
73 \( 1 + (588. - 588. i)T - 3.89e5iT^{2} \)
79 \( 1 - 693. iT - 4.93e5T^{2} \)
83 \( 1 + (735. + 735. i)T + 5.71e5iT^{2} \)
89 \( 1 - 755.T + 7.04e5T^{2} \)
97 \( 1 + (760. + 760. i)T + 9.12e5iT^{2} \)
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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

−12.68661014565850994704078945593, −11.51119443901289550227031774918, −10.42364970445023077209799525371, −9.528191057762947634107358335444, −8.451111268655364727766346337073, −7.34545932311140986409975478098, −5.64080693879029081583454343302, −4.12078189837982770078230481901, −3.34617403768294319892882250485, −0.27242399169493834737290162625, 2.31501779468854748290036924502, 3.53544682960389209331405551847, 5.71833837588931063526625267584, 6.93616187571430855995880258631, 7.71899680299683071876448862046, 9.018311643140293079122654197034, 10.08788023167173082771784977776, 11.66125122980018820718647652284, 12.42445194314424075055329020897, 13.00852160971514639590434839113

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