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Properties

Label 2-360-9.7-c1-0-5
Degree $2$
Conductor $360$
Sign $0.766 + 0.642i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (−1.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (1.5 + 2.59i)9-s + (2.5 − 4.33i)11-s − 1.73i·15-s + 3·17-s + 5·19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s − 5.19i·27-s + (5 − 8.66i)29-s + (1 + 1.73i)31-s + (−7.5 + 4.33i)33-s + 4·37-s + (1.5 + 2.59i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (0.223 + 0.387i)5-s + (0.5 + 0.866i)9-s + (0.753 − 1.30i)11-s − 0.447i·15-s + 0.727·17-s + 1.14·19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s − 0.999i·27-s + (0.928 − 1.60i)29-s + (0.179 + 0.311i)31-s + (−1.30 + 0.753i)33-s + 0.657·37-s + (0.234 + 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04752 - 0.381267i\)
\(L(\frac12)\) \(\approx\) \(1.04752 - 0.381267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.55075314164370822668871363263, −10.52907382032325495535602535526, −9.717616971287628595149477259521, −8.388381885303638403986615964534, −7.47212695702300770696170057480, −6.26662905851962067260663259285, −5.85499433009471847621260909731, −4.42459630625414077181508781554, −2.88962526442474332077175319755, −1.04346881382601896657000877217, 1.41928798145466109965702855674, 3.57323625325750740051187284558, 4.74185505492159377343008750331, 5.53803659567040943831769290859, 6.68297812665424680765690603531, 7.61921876845553442276071254155, 9.135287886110341675777342137750, 9.751780095792664076777605127566, 10.49731634854431653154893426223, 11.86779269749615893667375109294

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