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Properties

Label 2-2160-9.7-c1-0-3
Degree $2$
Conductor $2160$
Sign $-0.939 + 0.342i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)5-s + (−2 + 3.46i)7-s + (−1.5 + 2.59i)11-s + (2 + 3.46i)13-s − 3·17-s − 5·19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + (3 − 5.19i)29-s + (1 + 1.73i)31-s − 3.99·35-s − 4·37-s + (−1.5 − 2.59i)41-s + (5.5 − 9.52i)43-s + (−4.49 − 7.79i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.755 + 1.30i)7-s + (−0.452 + 0.783i)11-s + (0.554 + 0.960i)13-s − 0.727·17-s − 1.14·19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + (0.557 − 0.964i)29-s + (0.179 + 0.311i)31-s − 0.676·35-s − 0.657·37-s + (−0.234 − 0.405i)41-s + (0.838 − 1.45i)43-s + (−0.642 − 1.11i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4565246712\)
\(L(\frac12)\) \(\approx\) \(0.4565246712\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.402942577425087398479176564554, −8.793346068742603410345359597751, −8.161472199722796633941396541098, −6.83615687115414368088805621064, −6.48961617953921916233822252534, −5.71818996371537310561122043953, −4.66631841423249112835380426293, −3.80749145910028712798896568077, −2.44228577604610559671111968615, −2.14067710149733722149566210170, 0.15602568462835910984476964763, 1.32105472799674481283707591934, 2.84556844312553614716460414787, 3.68415164302403484624035978709, 4.48322592089821071348484888830, 5.57058215124847295624119353187, 6.28768471629385498977873769095, 7.03183235798077611270013236439, 8.001858361789386089999430210507, 8.515108316906787822115100080077

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