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Properties

Label 2-1620-9.4-c3-0-25
Degree $2$
Conductor $1620$
Sign $0.766 - 0.642i$
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 + (9.81 + 17.0i)7-s + (14.5 + 25.2i)11-s + (6.81 − 11.8i)13-s + 39.0·17-s + 43.7·19-s + (79.6 − 137. i)23-s + (−12.5 − 21.6i)25-s + (−12.5 − 21.7i)29-s + (58.3 − 101. i)31-s − 98.1·35-s + 329.·37-s + (90.4 − 156. i)41-s + (106. + 185. i)43-s + (−89.7 − 155. i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.530 + 0.918i)7-s + (0.399 + 0.692i)11-s + (0.145 − 0.251i)13-s + 0.557·17-s + 0.528·19-s + (0.721 − 1.25i)23-s + (−0.100 − 0.173i)25-s + (−0.0805 − 0.139i)29-s + (0.338 − 0.585i)31-s − 0.474·35-s + 1.46·37-s + (0.344 − 0.596i)41-s + (0.378 + 0.656i)43-s + (−0.278 − 0.482i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
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: \((2,\ 1620,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.548730662\)
\(L(\frac12)\) \(\approx\) \(2.548730662\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.81 - 17.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-14.5 - 25.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.81 + 11.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 39.0T + 4.91e3T^{2} \)
19 \( 1 - 43.7T + 6.85e3T^{2} \)
23 \( 1 + (-79.6 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (12.5 + 21.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-58.3 + 101. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 329.T + 5.06e4T^{2} \)
41 \( 1 + (-90.4 + 156. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-106. - 185. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (89.7 + 155. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 60.8T + 1.48e5T^{2} \)
59 \( 1 + (191. - 331. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (216. + 375. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (62.5 - 108. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 30.0T + 3.57e5T^{2} \)
73 \( 1 - 676.T + 3.89e5T^{2} \)
79 \( 1 + (-416. - 722. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (176. + 306. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 948.T + 7.04e5T^{2} \)
97 \( 1 + (-142. - 247. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.160364782166852053144839177286, −8.245684757701369712552108230988, −7.63370616476337396337014726218, −6.69072067811630635962144418993, −5.88314402104634907256565536079, −4.99236831405856136875597776901, −4.14745633133074841808396199661, −2.96500203868896805043236545187, −2.15342811453189773978870461046, −0.861753327612376374380857054479, 0.796014169462326223400134773247, 1.44184925562258463136614289839, 3.06388458548835328083139290578, 3.88975195854283117465843104705, 4.74869631110789855131657499898, 5.60864743924526375339004376301, 6.56932322672263497957855961105, 7.56067770387060485822742857266, 7.962198482891614458847754191787, 9.018895847286216496214744378293

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