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Properties

Label 2-180-45.22-c2-0-2
Degree $2$
Conductor $180$
Sign $-0.526 - 0.850i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
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  + (0.631 + 2.93i)3-s + (2.17 + 4.50i)5-s + (−1.60 + 5.98i)7-s + (−8.20 + 3.70i)9-s + (−6.53 − 11.3i)11-s + (−2.42 − 9.03i)13-s + (−11.8 + 9.21i)15-s + (19.2 + 19.2i)17-s + 18.2i·19-s + (−18.5 − 0.920i)21-s + (−2.47 − 9.22i)23-s + (−15.5 + 19.5i)25-s + (−16.0 − 21.7i)27-s + (16.8 − 9.72i)29-s + (−16.5 + 28.5i)31-s + ⋯
L(s)  = 1  + (0.210 + 0.977i)3-s + (0.434 + 0.900i)5-s + (−0.228 + 0.854i)7-s + (−0.911 + 0.411i)9-s + (−0.593 − 1.02i)11-s + (−0.186 − 0.695i)13-s + (−0.789 + 0.614i)15-s + (1.13 + 1.13i)17-s + 0.961i·19-s + (−0.883 − 0.0438i)21-s + (−0.107 − 0.401i)23-s + (−0.623 + 0.782i)25-s + (−0.594 − 0.804i)27-s + (0.580 − 0.335i)29-s + (−0.532 + 0.922i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.711958 + 1.27812i\)
\(L(\frac12)\) \(\approx\) \(0.711958 + 1.27812i\)
\(L(2)\) 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 \( 1 + (-0.631 - 2.93i)T \)
5 \( 1 + (-2.17 - 4.50i)T \)
good7 \( 1 + (1.60 - 5.98i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (6.53 + 11.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.42 + 9.03i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-19.2 - 19.2i)T + 289iT^{2} \)
19 \( 1 - 18.2iT - 361T^{2} \)
23 \( 1 + (2.47 + 9.22i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (-16.8 + 9.72i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16.5 - 28.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-48.9 - 48.9i)T + 1.36e3iT^{2} \)
41 \( 1 + (-22.2 + 38.5i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.3 - 6.00i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-14.7 + 54.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-33.8 + 33.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-59.5 - 34.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (16.7 + 29.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.7 - 4.49i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 27.5T + 5.04e3T^{2} \)
73 \( 1 + (-58.6 + 58.6i)T - 5.32e3iT^{2} \)
79 \( 1 + (121. - 69.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (40.2 + 10.7i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 58.9iT - 7.92e3T^{2} \)
97 \( 1 + (-35.9 + 134. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.76432971978257599348197182443, −11.59645505661185395892090155338, −10.39281435922636592285568725399, −10.14818258527609947095764106052, −8.751525181843014520903210777731, −7.902917621566940280794704809159, −6.02914068101435089887885432820, −5.50763878578761514595218210554, −3.58601825875703391383922745734, −2.65735161323103259375491734630, 0.877015705408865635481356688770, 2.47488578422964824475784797734, 4.43811764074418788488347478917, 5.70524131125693469595473618797, 7.14032425196927125003548265306, 7.70305993063958839655902782039, 9.155440790214131054045530891195, 9.829393698917546773574517814166, 11.33249974481976262829739708080, 12.36460218490917676700985102581

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