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Properties

Label 2-180-12.11-c7-0-42
Degree $2$
Conductor $180$
Sign $-0.987 + 0.155i$
Analytic cond. $56.2293$
Root an. cond. $7.49862$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 2.57i)2-s + (114. + 56.8i)4-s + 125i·5-s − 118. i·7-s + (−1.11e3 − 921. i)8-s + (322. − 1.37e3i)10-s + 4.55e3·11-s − 7.99e3·13-s + (−304. + 1.30e3i)14-s + (9.93e3 + 1.30e4i)16-s − 2.49e4i·17-s + 4.89e4i·19-s + (−7.10e3 + 1.43e4i)20-s + (−5.01e4 − 1.17e4i)22-s − 4.76e4·23-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)2-s + (0.896 + 0.443i)4-s + 0.447i·5-s − 0.130i·7-s + (−0.771 − 0.636i)8-s + (0.101 − 0.435i)10-s + 1.03·11-s − 1.00·13-s + (−0.0296 + 0.126i)14-s + (0.606 + 0.795i)16-s − 1.23i·17-s + 1.63i·19-s + (−0.198 + 0.400i)20-s + (−1.00 − 0.234i)22-s − 0.817·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.987 + 0.155i$
Analytic conductor: \(56.2293\)
Root analytic conductor: \(7.49862\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :7/2),\ -0.987 + 0.155i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.04663419324\)
\(L(\frac12)\) \(\approx\) \(0.04663419324\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.0 + 2.57i)T \)
3 \( 1 \)
5 \( 1 - 125iT \)
good7 \( 1 + 118. iT - 8.23e5T^{2} \)
11 \( 1 - 4.55e3T + 1.94e7T^{2} \)
13 \( 1 + 7.99e3T + 6.27e7T^{2} \)
17 \( 1 + 2.49e4iT - 4.10e8T^{2} \)
19 \( 1 - 4.89e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.76e4T + 3.40e9T^{2} \)
29 \( 1 - 4.10e4iT - 1.72e10T^{2} \)
31 \( 1 - 4.00e4iT - 2.75e10T^{2} \)
37 \( 1 - 1.40e5T + 9.49e10T^{2} \)
41 \( 1 + 5.59e5iT - 1.94e11T^{2} \)
43 \( 1 - 3.43e5iT - 2.71e11T^{2} \)
47 \( 1 - 9.94e5T + 5.06e11T^{2} \)
53 \( 1 + 2.02e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.65e6T + 2.48e12T^{2} \)
61 \( 1 + 3.11e6T + 3.14e12T^{2} \)
67 \( 1 - 3.95e5iT - 6.06e12T^{2} \)
71 \( 1 - 1.79e6T + 9.09e12T^{2} \)
73 \( 1 + 3.77e6T + 1.10e13T^{2} \)
79 \( 1 + 1.67e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.39e6T + 2.71e13T^{2} \)
89 \( 1 + 9.29e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.46e7T + 8.07e13T^{2} \)
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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

−10.69601220812599987736946296128, −9.853179674711580000354658348403, −9.063224967719035402225982357005, −7.76316302127583736562614366865, −7.02850610009006549060474922550, −5.87229595742700525396283523350, −4.01855065809980818408641957320, −2.72983129312219138613601191666, −1.46845768955592440728817722663, −0.01708404097424627046069586933, 1.27731017005313270205268436279, 2.53196874403349561682655094770, 4.33754190343987182940683311907, 5.78478060777643934422749076254, 6.80813117489597635817375206690, 7.87929685938491866915536401739, 8.929013983098981719691745487663, 9.588146416942557531049867622562, 10.69294123532946662672380154780, 11.73279483211647085845288176252

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