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Properties

Label 2-405-9.4-c3-0-11
Degree $2$
Conductor $405$
Sign $0.939 + 0.342i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
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  + (−2.22 − 3.86i)2-s + (−5.94 + 10.2i)4-s + (−2.5 + 4.33i)5-s + (−2.54 − 4.40i)7-s + 17.3·8-s + 22.2·10-s + (−29.1 − 50.4i)11-s + (−10.6 + 18.3i)13-s + (−11.3 + 19.6i)14-s + (8.90 + 15.4i)16-s − 68.8·17-s − 40.8·19-s + (−29.7 − 51.4i)20-s + (−129. + 225. i)22-s + (72.1 − 124. i)23-s + ⋯
L(s)  = 1  + (−0.788 − 1.36i)2-s + (−0.742 + 1.28i)4-s + (−0.223 + 0.387i)5-s + (−0.137 − 0.237i)7-s + 0.765·8-s + 0.705·10-s + (−0.799 − 1.38i)11-s + (−0.226 + 0.391i)13-s + (−0.216 + 0.374i)14-s + (0.139 + 0.240i)16-s − 0.982·17-s − 0.492·19-s + (−0.332 − 0.575i)20-s + (−1.25 + 2.18i)22-s + (0.654 − 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5896289839\)
\(L(\frac12)\) \(\approx\) \(0.5896289839\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (2.22 + 3.86i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (2.54 + 4.40i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (29.1 + 50.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (10.6 - 18.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 68.8T + 4.91e3T^{2} \)
19 \( 1 + 40.8T + 6.85e3T^{2} \)
23 \( 1 + (-72.1 + 124. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-110. - 190. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (145. - 252. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 260.T + 5.06e4T^{2} \)
41 \( 1 + (-84.8 + 147. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-219. - 379. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-127. - 221. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 214.T + 1.48e5T^{2} \)
59 \( 1 + (165. - 287. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (27.4 + 47.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (379. - 656. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 904.T + 3.57e5T^{2} \)
73 \( 1 - 866.T + 3.89e5T^{2} \)
79 \( 1 + (103. + 179. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-231. - 401. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 601.T + 7.04e5T^{2} \)
97 \( 1 + (114. + 198. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.89207347893501035291344015938, −10.20979905923025584670132490903, −8.893490567373274496778925703057, −8.572397879540337764853315652655, −7.25792602470848503723650397606, −6.10125004918957331032747129695, −4.51179607368634170085823196407, −3.23966125061973645538537474728, −2.46789062674345609632864465558, −0.853898012077550467950396655539, 0.35138344149996091281616852386, 2.34458105192506310838462730947, 4.35194235417248260556491268150, 5.34793335035644911759527271075, 6.30625289906292482640812484449, 7.46266335059419804938787486746, 7.81066131195977962790529685977, 9.009964093967077707023370289320, 9.561882004322176378724129478935, 10.53237088272790888060666092224

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