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Properties

Label 2-360-120.59-c3-0-15
Degree $2$
Conductor $360$
Sign $0.488 - 0.872i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
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  + (1.51 − 2.38i)2-s + (−3.38 − 7.24i)4-s + (6.07 + 9.38i)5-s − 14.2·7-s + (−22.4 − 2.93i)8-s + (31.6 − 0.252i)10-s + 27.1i·11-s + 26.3·13-s + (−21.6 + 34.0i)14-s + (−41.0 + 49.0i)16-s − 85.4·17-s − 42.9·19-s + (47.4 − 75.8i)20-s + (64.7 + 41.2i)22-s + 192. i·23-s + ⋯
L(s)  = 1  + (0.537 − 0.843i)2-s + (−0.423 − 0.906i)4-s + (0.543 + 0.839i)5-s − 0.769·7-s + (−0.991 − 0.129i)8-s + (0.999 − 0.00799i)10-s + 0.743i·11-s + 0.562·13-s + (−0.413 + 0.649i)14-s + (−0.641 + 0.766i)16-s − 1.21·17-s − 0.519·19-s + (0.530 − 0.847i)20-s + (0.627 + 0.399i)22-s + 1.74i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.376500196\)
\(L(\frac12)\) \(\approx\) \(1.376500196\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.51 + 2.38i)T \)
3 \( 1 \)
5 \( 1 + (-6.07 - 9.38i)T \)
good7 \( 1 + 14.2T + 343T^{2} \)
11 \( 1 - 27.1iT - 1.33e3T^{2} \)
13 \( 1 - 26.3T + 2.19e3T^{2} \)
17 \( 1 + 85.4T + 4.91e3T^{2} \)
19 \( 1 + 42.9T + 6.85e3T^{2} \)
23 \( 1 - 192. iT - 1.21e4T^{2} \)
29 \( 1 - 237.T + 2.43e4T^{2} \)
31 \( 1 - 232. iT - 2.97e4T^{2} \)
37 \( 1 + 214.T + 5.06e4T^{2} \)
41 \( 1 + 427. iT - 6.89e4T^{2} \)
43 \( 1 - 317. iT - 7.95e4T^{2} \)
47 \( 1 + 78.8iT - 1.03e5T^{2} \)
53 \( 1 + 49.0iT - 1.48e5T^{2} \)
59 \( 1 - 386. iT - 2.05e5T^{2} \)
61 \( 1 + 11.0iT - 2.26e5T^{2} \)
67 \( 1 + 275. iT - 3.00e5T^{2} \)
71 \( 1 - 965.T + 3.57e5T^{2} \)
73 \( 1 - 816. iT - 3.89e5T^{2} \)
79 \( 1 + 332. iT - 4.93e5T^{2} \)
83 \( 1 + 459.T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3iT - 7.04e5T^{2} \)
97 \( 1 - 329. iT - 9.12e5T^{2} \)
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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

−11.07546464391804667633102415931, −10.38049926068483420628705888658, −9.639667228509401758132039557667, −8.757174762549464490408388679966, −6.96861447761502771251788994760, −6.33672579417361209105390714493, −5.18359593767382341915153534636, −3.85891003173662880470861666881, −2.85011439545152907087506596928, −1.68854170066283926129835512571, 0.38243176294801872428418662083, 2.59461216192527859799163060525, 4.03425378108981913070789647091, 4.96723417923710375544706739846, 6.27166972015094539971957495869, 6.51818788780137369820255826919, 8.269878934254587898665152099624, 8.689919623953637838738253209439, 9.676745442306139121749135154677, 10.89942879393218510188452196843

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