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Properties

Label 2-180-180.59-c1-0-16
Degree $2$
Conductor $180$
Sign $0.915 - 0.403i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (1.38 − 0.282i)2-s + (0.959 + 1.44i)3-s + (1.84 − 0.782i)4-s + (−1.62 + 1.53i)5-s + (1.73 + 1.72i)6-s + (−0.550 − 0.953i)7-s + (2.32 − 1.60i)8-s + (−1.15 + 2.76i)9-s + (−1.81 + 2.58i)10-s + (−2.84 − 4.92i)11-s + (2.89 + 1.90i)12-s + (2.07 + 1.19i)13-s + (−1.03 − 1.16i)14-s + (−3.77 − 0.864i)15-s + (2.77 − 2.88i)16-s − 1.29·17-s + ⋯
L(s)  = 1  + (0.979 − 0.199i)2-s + (0.554 + 0.832i)3-s + (0.920 − 0.391i)4-s + (−0.725 + 0.687i)5-s + (0.709 + 0.705i)6-s + (−0.208 − 0.360i)7-s + (0.823 − 0.567i)8-s + (−0.385 + 0.922i)9-s + (−0.574 + 0.818i)10-s + (−0.856 − 1.48i)11-s + (0.835 + 0.549i)12-s + (0.575 + 0.332i)13-s + (−0.275 − 0.311i)14-s + (−0.974 − 0.223i)15-s + (0.693 − 0.720i)16-s − 0.313·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.915 - 0.403i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.915 - 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00791 + 0.422925i\)
\(L(\frac12)\) \(\approx\) \(2.00791 + 0.422925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.282i)T \)
3 \( 1 + (-0.959 - 1.44i)T \)
5 \( 1 + (1.62 - 1.53i)T \)
good7 \( 1 + (0.550 + 0.953i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.84 + 4.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.07 - 1.19i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (-1.82 - 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.51 - 3.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.78 - 1.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.51iT - 37T^{2} \)
41 \( 1 + (-6.35 - 3.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.25 + 3.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.09 - 4.67i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + (3.66 - 6.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.48 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.85 - 3.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 9.86iT - 73T^{2} \)
79 \( 1 + (0.975 - 0.563i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.98 + 5.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.16iT - 89T^{2} \)
97 \( 1 + (-11.1 + 6.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.06242542002782130219925641367, −11.30534774776334373937049235468, −11.10110531369703027756036060975, −10.11393539740837651974796873167, −8.614909612114184703564811587742, −7.51567444162815480578790775489, −6.22431145103770003703000962531, −4.85884350452008456476648137955, −3.61976267040721083524794911079, −2.90498887668807598076223834389, 2.12083145093735768371966874183, 3.63677052791699701553990563672, 4.94821735878169180748002785113, 6.25864687639693842051458016525, 7.53821459798216483587432504531, 8.039590174019610060942642982852, 9.400386401963009016099929301707, 11.02697870971605749213957465552, 12.15213281329560894255844875582, 12.73567091310778125337552772458

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