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Properties

Label 2-120-3.2-c2-0-0
Degree $2$
Conductor $120$
Sign $-0.800 - 0.599i$
Analytic cond. $3.26976$
Root an. cond. $1.80824$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.40 − 1.79i)3-s + 2.23i·5-s − 10.2·7-s + (2.53 + 8.63i)9-s + 8.19i·11-s − 13.5·13-s + (4.02 − 5.36i)15-s + 15.4i·17-s − 25.4·19-s + (24.5 + 18.3i)21-s − 17.9i·23-s − 5.00·25-s + (9.44 − 25.2i)27-s − 42.0i·29-s + 38.4·31-s + ⋯
L(s)  = 1  + (−0.800 − 0.599i)3-s + 0.447i·5-s − 1.45·7-s + (0.281 + 0.959i)9-s + 0.744i·11-s − 1.04·13-s + (0.268 − 0.357i)15-s + 0.910i·17-s − 1.34·19-s + (1.16 + 0.874i)21-s − 0.778i·23-s − 0.200·25-s + (0.349 − 0.936i)27-s − 1.44i·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(3.26976\)
Root analytic conductor: \(1.80824\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1),\ -0.800 - 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0683915 + 0.205441i\)
\(L(\frac12)\) \(\approx\) \(0.0683915 + 0.205441i\)
\(L(2)\) 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 \( 1 + (2.40 + 1.79i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 + 10.2T + 49T^{2} \)
11 \( 1 - 8.19iT - 121T^{2} \)
13 \( 1 + 13.5T + 169T^{2} \)
17 \( 1 - 15.4iT - 289T^{2} \)
19 \( 1 + 25.4T + 361T^{2} \)
23 \( 1 + 17.9iT - 529T^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 - 38.4T + 961T^{2} \)
37 \( 1 - 11.8T + 1.36e3T^{2} \)
41 \( 1 - 46.3iT - 1.68e3T^{2} \)
43 \( 1 + 54.0T + 1.84e3T^{2} \)
47 \( 1 - 43.0iT - 2.20e3T^{2} \)
53 \( 1 + 82.7iT - 2.80e3T^{2} \)
59 \( 1 - 45.8iT - 3.48e3T^{2} \)
61 \( 1 + 93.6T + 3.72e3T^{2} \)
67 \( 1 - 34.4T + 4.48e3T^{2} \)
71 \( 1 - 68.0iT - 5.04e3T^{2} \)
73 \( 1 + 44.7T + 5.32e3T^{2} \)
79 \( 1 + 11.7T + 6.24e3T^{2} \)
83 \( 1 - 144. iT - 6.88e3T^{2} \)
89 \( 1 + 63.7iT - 7.92e3T^{2} \)
97 \( 1 - 63.9T + 9.40e3T^{2} \)
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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

−13.22456567201842075008807913101, −12.68210290285211321821521013883, −11.78290804101972411364445606251, −10.39678200899221445957525975574, −9.817762190154395370146637686614, −8.033200579063288274152280865514, −6.71946596762654416533419749157, −6.22002224173348558138485287582, −4.45800723735361511691008941740, −2.46338764434779017300378449339, 0.15687545200369335686533508709, 3.25976276116511647196398552708, 4.76590345747011203658606418902, 5.98881275739973653913262717555, 7.02016384908321371249350529119, 8.878350884526434233033059429113, 9.733362144214630948088554489550, 10.62837531945560615382159443674, 11.91472285695696778835369471500, 12.63205850142673952720161931245

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