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Properties

Label 2-405-45.14-c2-0-32
Degree $2$
Conductor $405$
Sign $0.206 + 0.978i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.577 − 1.00i)2-s + (1.33 − 2.30i)4-s + (1.45 + 4.78i)5-s + (6.09 − 3.51i)7-s − 7.70·8-s + (3.94 − 4.22i)10-s + (4.54 − 2.62i)11-s + (−15.8 − 9.17i)13-s + (−7.04 − 4.06i)14-s + (−0.878 − 1.52i)16-s + 24.8·17-s + 25.3·19-s + (12.9 + 3.00i)20-s + (−5.25 − 3.03i)22-s + (−5.44 + 9.42i)23-s + ⋯
L(s)  = 1  + (−0.288 − 0.500i)2-s + (0.333 − 0.576i)4-s + (0.291 + 0.956i)5-s + (0.870 − 0.502i)7-s − 0.962·8-s + (0.394 − 0.422i)10-s + (0.413 − 0.238i)11-s + (−1.22 − 0.706i)13-s + (−0.503 − 0.290i)14-s + (−0.0549 − 0.0951i)16-s + 1.46·17-s + 1.33·19-s + (0.648 + 0.150i)20-s + (−0.238 − 0.137i)22-s + (−0.236 + 0.409i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.206 + 0.978i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.206 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.38311 - 1.12115i\)
\(L(\frac12)\) \(\approx\) \(1.38311 - 1.12115i\)
\(L(2)\) 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 \( 1 + (-1.45 - 4.78i)T \)
good2 \( 1 + (0.577 + 1.00i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (-6.09 + 3.51i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-4.54 + 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (15.8 + 9.17i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 24.8T + 289T^{2} \)
19 \( 1 - 25.3T + 361T^{2} \)
23 \( 1 + (5.44 - 9.42i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.0 + 13.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-20.5 + 35.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 28.5iT - 1.36e3T^{2} \)
41 \( 1 + (37.0 + 21.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-41.0 + 23.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (15.1 + 26.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 1.12T + 2.80e3T^{2} \)
59 \( 1 + (44.2 + 25.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-39.1 - 67.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (18.3 + 10.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 14.4iT - 5.04e3T^{2} \)
73 \( 1 + 112. iT - 5.32e3T^{2} \)
79 \( 1 + (-8.96 - 15.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-58.3 - 101. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 18.0iT - 7.92e3T^{2} \)
97 \( 1 + (-0.442 + 0.255i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.69735700466485508594815843737, −10.06455901588973288812800387618, −9.490336747550517774558946012746, −7.86424164481508763021275054213, −7.25859098592104815478026536741, −5.98842461646309706714320246351, −5.17016092820688243297287541996, −3.44222690935911244424817093749, −2.34598824237185820523084819362, −0.949956463167870247499354973348, 1.47026448473469497688930665638, 2.96025018706459746721788536393, 4.62527765063307707699471800667, 5.41226917895940988562553305122, 6.66322933566558086857913943060, 7.70968421242939891810593024956, 8.344533963387018206372881399672, 9.266945169657735065034133500783, 10.00718546209186101612973413696, 11.71304430045251470440433746037

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