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Properties

Label 2-180-4.3-c6-0-42
Degree $2$
Conductor $180$
Sign $-0.996 + 0.0829i$
Analytic cond. $41.4097$
Root an. cond. $6.43503$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.331 − 7.99i)2-s + (−63.7 + 5.30i)4-s + 55.9·5-s + 552. i·7-s + (63.5 + 508. i)8-s + (−18.5 − 446. i)10-s + 1.28e3i·11-s − 2.51e3·13-s + (4.41e3 − 183. i)14-s + (4.03e3 − 676. i)16-s − 4.59e3·17-s − 9.80e3i·19-s + (−3.56e3 + 296. i)20-s + (1.02e4 − 426. i)22-s − 1.52e4i·23-s + ⋯
L(s)  = 1  + (−0.0414 − 0.999i)2-s + (−0.996 + 0.0829i)4-s + 0.447·5-s + 1.60i·7-s + (0.124 + 0.992i)8-s + (−0.0185 − 0.446i)10-s + 0.965i·11-s − 1.14·13-s + (1.60 − 0.0667i)14-s + (0.986 − 0.165i)16-s − 0.935·17-s − 1.42i·19-s + (−0.445 + 0.0370i)20-s + (0.964 − 0.0400i)22-s − 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0829i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.996 + 0.0829i$
Analytic conductor: \(41.4097\)
Root analytic conductor: \(6.43503\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :3),\ -0.996 + 0.0829i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4918880351\)
\(L(\frac12)\) \(\approx\) \(0.4918880351\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.331 + 7.99i)T \)
3 \( 1 \)
5 \( 1 - 55.9T \)
good7 \( 1 - 552. iT - 1.17e5T^{2} \)
11 \( 1 - 1.28e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.51e3T + 4.82e6T^{2} \)
17 \( 1 + 4.59e3T + 2.41e7T^{2} \)
19 \( 1 + 9.80e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.52e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.43e4T + 5.94e8T^{2} \)
31 \( 1 + 3.44e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.14e4T + 2.56e9T^{2} \)
41 \( 1 - 3.76e4T + 4.75e9T^{2} \)
43 \( 1 + 1.53e3iT - 6.32e9T^{2} \)
47 \( 1 + 1.31e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.28e5T + 2.21e10T^{2} \)
59 \( 1 + 2.71e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.52e5T + 5.15e10T^{2} \)
67 \( 1 + 7.17e4iT - 9.04e10T^{2} \)
71 \( 1 - 1.93e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.25e5T + 1.51e11T^{2} \)
79 \( 1 - 6.09e5iT - 2.43e11T^{2} \)
83 \( 1 - 5.92e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.54e5T + 4.96e11T^{2} \)
97 \( 1 + 1.46e6T + 8.32e11T^{2} \)
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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

−11.18579291211447934327828917587, −10.00342598624549626193104325207, −9.229093454967639842321667677754, −8.492561625641401116506727731936, −6.84117434966063597286198262729, −5.34167696787587725604973218288, −4.54802594375155182030211737008, −2.53126207628334194652835346368, −2.19956332938152115509407297180, −0.14737194317853814818079906570, 1.21608680609291343984198053208, 3.50229015731722834443895208929, 4.64057194272119529286751377526, 5.84354858436625470996352015923, 6.96888804755442014510566387515, 7.71252887845843706990885768790, 8.889498792075824399007718982217, 10.02496355799948910729299937703, 10.67721181895801597279070216337, 12.26270353127708440395534202920

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