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Properties

Label 2-180-15.14-c2-0-2
Degree $2$
Conductor $180$
Sign $0.784 + 0.619i$
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  + (4.79 − 1.41i)5-s − 6.78i·7-s + 9.89i·11-s − 20.3i·13-s + 19.1·17-s + 12·19-s + 9.59·23-s + (20.9 − 13.5i)25-s − 8.48i·29-s − 38·31-s + (−9.59 − 32.5i)35-s − 6.78i·37-s + 69.2i·41-s + 67.8i·43-s − 76.7·47-s + ⋯
L(s)  = 1  + (0.959 − 0.282i)5-s − 0.968i·7-s + 0.899i·11-s − 1.56i·13-s + 1.12·17-s + 0.631·19-s + 0.417·23-s + (0.839 − 0.542i)25-s − 0.292i·29-s − 1.22·31-s + (−0.274 − 0.929i)35-s − 0.183i·37-s + 1.69i·41-s + 1.57i·43-s − 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.784 + 0.619i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.784 + 0.619i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64290 - 0.570603i\)
\(L(\frac12)\) \(\approx\) \(1.64290 - 0.570603i\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-4.79 + 1.41i)T \)
good7 \( 1 + 6.78iT - 49T^{2} \)
11 \( 1 - 9.89iT - 121T^{2} \)
13 \( 1 + 20.3iT - 169T^{2} \)
17 \( 1 - 19.1T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 - 9.59T + 529T^{2} \)
29 \( 1 + 8.48iT - 841T^{2} \)
31 \( 1 + 38T + 961T^{2} \)
37 \( 1 + 6.78iT - 1.36e3T^{2} \)
41 \( 1 - 69.2iT - 1.68e3T^{2} \)
43 \( 1 - 67.8iT - 1.84e3T^{2} \)
47 \( 1 + 76.7T + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 83.4iT - 3.48e3T^{2} \)
61 \( 1 + 70T + 3.72e3T^{2} \)
67 \( 1 - 108. iT - 4.48e3T^{2} \)
71 \( 1 - 118. iT - 5.04e3T^{2} \)
73 \( 1 + 13.5iT - 5.32e3T^{2} \)
79 \( 1 - 30T + 6.24e3T^{2} \)
83 \( 1 + 134.T + 6.88e3T^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + 94.9iT - 9.40e3T^{2} \)
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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

−12.71698066420954318851247604560, −11.20404488560958383587438803612, −10.04979135040049408856054909340, −9.720381193370290913586798042258, −8.113062311906171487269760935515, −7.20172289832273122313946574996, −5.82493972515776846081865023486, −4.80645800003368975649095218311, −3.13297100361280610857479476312, −1.22985466817530681602469551786, 1.84852352926706722258236906010, 3.31998037859101341268502773024, 5.24388255188249666489574111502, 6.05184422118524413711327330613, 7.22413730712253462638454324193, 8.821071895576845704663531337556, 9.339605546805963659813582799435, 10.54886274406848368639598274773, 11.60083604555118939964655992469, 12.44506649711823621424567703189

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