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Properties

Label 2-180-5.4-c5-0-4
Degree $2$
Conductor $180$
Sign $0.715 - 0.698i$
Analytic cond. $28.8690$
Root an. cond. $5.37299$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (40 − 39.0i)5-s + 124. i·7-s + 80·11-s + 374. i·13-s + 546. i·17-s + 12·19-s + 2.03e3i·23-s + (75.0 − 3.12e3i)25-s + 4.56e3·29-s − 344·31-s + (4.87e3 + 4.99e3i)35-s − 4.37e3i·37-s + 1.42e4·41-s + 1.89e4i·43-s + 2.45e4i·47-s + ⋯
L(s)  = 1  + (0.715 − 0.698i)5-s + 0.963i·7-s + 0.199·11-s + 0.615i·13-s + 0.458i·17-s + 0.00762·19-s + 0.800i·23-s + (0.0240 − 0.999i)25-s + 1.00·29-s − 0.0642·31-s + (0.673 + 0.689i)35-s − 0.525i·37-s + 1.32·41-s + 1.56i·43-s + 1.61i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(28.8690\)
Root analytic conductor: \(5.37299\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :5/2),\ 0.715 - 0.698i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.160153943\)
\(L(\frac12)\) \(\approx\) \(2.160153943\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 + (-40 + 39.0i)T \)
good7 \( 1 - 124. iT - 1.68e4T^{2} \)
11 \( 1 - 80T + 1.61e5T^{2} \)
13 \( 1 - 374. iT - 3.71e5T^{2} \)
17 \( 1 - 546. iT - 1.41e6T^{2} \)
19 \( 1 - 12T + 2.47e6T^{2} \)
23 \( 1 - 2.03e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.56e3T + 2.05e7T^{2} \)
31 \( 1 + 344T + 2.86e7T^{2} \)
37 \( 1 + 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.42e4T + 1.15e8T^{2} \)
43 \( 1 - 1.89e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.45e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.74e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.80e4T + 7.14e8T^{2} \)
61 \( 1 + 8.20e3T + 8.44e8T^{2} \)
67 \( 1 + 1.32e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.84e4T + 1.80e9T^{2} \)
73 \( 1 + 4.24e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.26e3T + 3.07e9T^{2} \)
83 \( 1 - 3.34e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.43e4T + 5.58e9T^{2} \)
97 \( 1 - 1.36e5iT - 8.58e9T^{2} \)
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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

−12.07367796741795029133552115982, −10.94736781371061494027060391980, −9.594456587427478983868552526716, −9.053417070806083120433221891367, −7.972213851543689589465888100099, −6.39789183050700820515667307099, −5.55606019748555247702478411566, −4.35626446000563198366405428020, −2.56509623474353642146251131365, −1.29828393778213581644091259787, 0.75695663963673858563771283106, 2.43059818381194239510965387159, 3.76481943805278164227286650819, 5.23173192909360231508550734281, 6.52086347672874051070813537218, 7.30904903336771331826035202470, 8.610290844469607963396679934722, 9.965768940438104500625352254883, 10.45624677069687431765481900585, 11.49353406032626824673228809992

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