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Properties

Label 2-405-9.4-c3-0-27
Degree $2$
Conductor $405$
Sign $-0.173 - 0.984i$
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  + (1.92 + 3.33i)2-s + (−3.41 + 5.91i)4-s + (2.5 − 4.33i)5-s + (13.1 + 22.8i)7-s + 4.52·8-s + 19.2·10-s + (2.53 + 4.39i)11-s + (41.3 − 71.6i)13-s + (−50.6 + 87.7i)14-s + (36.0 + 62.3i)16-s + 52.5·17-s − 29.8·19-s + (17.0 + 29.5i)20-s + (−9.77 + 16.9i)22-s + (49.1 − 85.1i)23-s + ⋯
L(s)  = 1  + (0.680 + 1.17i)2-s + (−0.426 + 0.738i)4-s + (0.223 − 0.387i)5-s + (0.710 + 1.23i)7-s + 0.199·8-s + 0.608·10-s + (0.0696 + 0.120i)11-s + (0.882 − 1.52i)13-s + (−0.967 + 1.67i)14-s + (0.562 + 0.974i)16-s + 0.750·17-s − 0.360·19-s + (0.190 + 0.330i)20-s + (−0.0947 + 0.164i)22-s + (0.445 − 0.771i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 405,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.486584679\)
\(L(\frac12)\) \(\approx\) \(3.486584679\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.92 - 3.33i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-13.1 - 22.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-2.53 - 4.39i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-41.3 + 71.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 52.5T + 4.91e3T^{2} \)
19 \( 1 + 29.8T + 6.85e3T^{2} \)
23 \( 1 + (-49.1 + 85.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-83.9 - 145. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (95.3 - 165. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 365.T + 5.06e4T^{2} \)
41 \( 1 + (55.8 - 96.7i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-201. - 349. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-116. - 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 410.T + 1.48e5T^{2} \)
59 \( 1 + (-76.0 + 131. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (266. + 460. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (306. - 531. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 413.T + 3.57e5T^{2} \)
73 \( 1 + 114.T + 3.89e5T^{2} \)
79 \( 1 + (39.5 + 68.5i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (713. + 1.23e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 450.T + 7.04e5T^{2} \)
97 \( 1 + (718. + 1.24e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.07382501863723961579322836036, −10.23705555441596841602969012239, −8.660853117388097551065833302907, −8.392193332571037973416730256380, −7.24299134190912976415145657165, −6.02859058872587507939314634020, −5.45589062313432219571450505213, −4.71387589198935980207559107384, −3.14814590930375470168181931129, −1.43178698246530459002536021264, 1.13061778395147925968531348964, 2.11844280633748984754206853028, 3.70208421303497943931356888372, 4.15396517368751670373484120546, 5.43317574337670770859854661888, 6.84387978895466757819976549888, 7.66649493390172475492038250987, 8.989057579863817091953366468073, 10.16097585013009424376298878599, 10.76187937837398580780816987411

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