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Properties

Label 2-1620-9.4-c3-0-46
Degree $2$
Conductor $1620$
Sign $-0.642 - 0.766i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (−11.2 − 19.4i)7-s + (−28.2 − 48.9i)11-s + (−21.7 + 37.7i)13-s + 34.9·17-s + 77.1·19-s + (−61.2 + 106. i)23-s + (−12.5 − 21.6i)25-s + (−136. − 236. i)29-s + (148. − 257. i)31-s − 112.·35-s − 267.·37-s + (90.5 − 156. i)41-s + (−184. − 320. i)43-s + (56.2 + 97.4i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.605 − 1.04i)7-s + (−0.774 − 1.34i)11-s + (−0.464 + 0.804i)13-s + 0.499·17-s + 0.931·19-s + (−0.555 + 0.961i)23-s + (−0.100 − 0.173i)25-s + (−0.874 − 1.51i)29-s + (0.862 − 1.49i)31-s − 0.541·35-s − 1.19·37-s + (0.344 − 0.597i)41-s + (−0.655 − 1.13i)43-s + (0.174 + 0.302i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3032741166\)
\(L(\frac12)\) \(\approx\) \(0.3032741166\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (11.2 + 19.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (28.2 + 48.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (21.7 - 37.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 34.9T + 4.91e3T^{2} \)
19 \( 1 - 77.1T + 6.85e3T^{2} \)
23 \( 1 + (61.2 - 106. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (136. + 236. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-148. + 257. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 267.T + 5.06e4T^{2} \)
41 \( 1 + (-90.5 + 156. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (184. + 320. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-56.2 - 97.4i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 23.1T + 1.48e5T^{2} \)
59 \( 1 + (139. - 241. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-196. - 340. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (197. - 341. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 973.T + 3.57e5T^{2} \)
73 \( 1 + 760.T + 3.89e5T^{2} \)
79 \( 1 + (-415. - 720. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-259. - 450. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (-419. - 727. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.451763834424315527238450878870, −7.69694053635162542200935645968, −7.03699288042229713139671973012, −5.91787410123056119987775351564, −5.41262050639847343334872775012, −4.14179251243000766831432586364, −3.49895696387847854833636676280, −2.32889328724062598884439289137, −0.952128080944348530282458022854, −0.07362713336006339033748468749, 1.66795341557183549504389111161, 2.73004835987637459920248613782, 3.28660458940055832709747496198, 4.89772095366702558187557925076, 5.32308338689490660874494669391, 6.34653207772872429993248068366, 7.13872300833524186304655800026, 7.891785919534658104967113604878, 8.793528904530396736138171016277, 9.722967841002460541681750884020

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