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Properties

Label 2-180-4.3-c8-0-36
Degree $2$
Conductor $180$
Sign $-0.292 - 0.956i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.51 + 12.8i)2-s + (−74.8 − 244. i)4-s + 279.·5-s + 2.23e3i·7-s + (3.86e3 + 1.36e3i)8-s + (−2.66e3 + 3.59e3i)10-s + 1.13e4i·11-s + 4.34e4·13-s + (−2.87e4 − 2.12e4i)14-s + (−5.43e4 + 3.66e4i)16-s + 1.06e5·17-s + 1.69e5i·19-s + (−2.09e4 − 6.84e4i)20-s + (−1.45e5 − 1.07e5i)22-s − 3.01e5i·23-s + ⋯
L(s)  = 1  + (−0.594 + 0.803i)2-s + (−0.292 − 0.956i)4-s + 0.447·5-s + 0.931i·7-s + (0.942 + 0.334i)8-s + (−0.266 + 0.359i)10-s + 0.774i·11-s + 1.52·13-s + (−0.748 − 0.554i)14-s + (−0.829 + 0.558i)16-s + 1.27·17-s + 1.29i·19-s + (−0.130 − 0.427i)20-s + (−0.622 − 0.460i)22-s − 1.07i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ -0.292 - 0.956i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.871728415\)
\(L(\frac12)\) \(\approx\) \(1.871728415\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.51 - 12.8i)T \)
3 \( 1 \)
5 \( 1 - 279.T \)
good7 \( 1 - 2.23e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.13e4iT - 2.14e8T^{2} \)
13 \( 1 - 4.34e4T + 8.15e8T^{2} \)
17 \( 1 - 1.06e5T + 6.97e9T^{2} \)
19 \( 1 - 1.69e5iT - 1.69e10T^{2} \)
23 \( 1 + 3.01e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.13e6T + 5.00e11T^{2} \)
31 \( 1 + 1.49e6iT - 8.52e11T^{2} \)
37 \( 1 - 3.00e6T + 3.51e12T^{2} \)
41 \( 1 - 1.45e6T + 7.98e12T^{2} \)
43 \( 1 + 1.84e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.23e6iT - 2.38e13T^{2} \)
53 \( 1 - 2.47e6T + 6.22e13T^{2} \)
59 \( 1 - 1.98e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.31e7T + 1.91e14T^{2} \)
67 \( 1 - 1.38e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.24e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.85e7T + 8.06e14T^{2} \)
79 \( 1 + 5.42e7iT - 1.51e15T^{2} \)
83 \( 1 - 8.59e7iT - 2.25e15T^{2} \)
89 \( 1 + 4.60e7T + 3.93e15T^{2} \)
97 \( 1 + 3.17e7T + 7.83e15T^{2} \)
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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

−11.26940426596770544441537146947, −10.14227610897972884127623619612, −9.376328579319432593544383893417, −8.409075741101567550445950353481, −7.50721326244645051591965204311, −5.98708707782965020482801798629, −5.73032830386349757224024808693, −4.08142702428272229260845926038, −2.19087945136590191624984495908, −1.05377355916774932371923326529, 0.69089534996386840595768637023, 1.43775044019363709478695459981, 3.10081122809096477543230989943, 3.92366058175319397851554345583, 5.52450375450149624183551980944, 6.95603612309709836933859875273, 8.023632931320462024011030564619, 9.030251333787582140747401558918, 9.938791801318879075802848968573, 10.98419456857290435033463407732

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