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Properties

Label 2-405-45.29-c2-0-20
Degree $2$
Conductor $405$
Sign $0.738 + 0.674i$
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  + (1.14 − 1.97i)2-s + (−0.608 − 1.05i)4-s + (−3.68 + 3.38i)5-s + (−2.98 − 1.72i)7-s + 6.35·8-s + (2.48 + 11.1i)10-s + (3.89 + 2.25i)11-s + (10.2 − 5.92i)13-s + (−6.81 + 3.93i)14-s + (9.69 − 16.7i)16-s + 23.3·17-s + 11.0·19-s + (5.80 + 1.82i)20-s + (8.90 − 5.14i)22-s + (14.9 + 25.8i)23-s + ⋯
L(s)  = 1  + (0.570 − 0.988i)2-s + (−0.152 − 0.263i)4-s + (−0.736 + 0.676i)5-s + (−0.426 − 0.246i)7-s + 0.794·8-s + (0.248 + 1.11i)10-s + (0.354 + 0.204i)11-s + (0.788 − 0.455i)13-s + (−0.486 + 0.281i)14-s + (0.605 − 1.04i)16-s + 1.37·17-s + 0.582·19-s + (0.290 + 0.0910i)20-s + (0.404 − 0.233i)22-s + (0.648 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.738 + 0.674i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.738 + 0.674i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.17529 - 0.844422i\)
\(L(\frac12)\) \(\approx\) \(2.17529 - 0.844422i\)
\(L(2)\) 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 \( 1 + (3.68 - 3.38i)T \)
good2 \( 1 + (-1.14 + 1.97i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (2.98 + 1.72i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.89 - 2.25i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.2 + 5.92i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 23.3T + 289T^{2} \)
19 \( 1 - 11.0T + 361T^{2} \)
23 \( 1 + (-14.9 - 25.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-30.8 - 17.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (15.1 + 26.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 5.11iT - 1.36e3T^{2} \)
41 \( 1 + (19.6 - 11.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-58.8 - 33.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (23.1 - 40.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 68.0T + 2.80e3T^{2} \)
59 \( 1 + (29.9 - 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8.68 - 15.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (85.1 - 49.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 134. iT - 5.04e3T^{2} \)
73 \( 1 + 134. iT - 5.32e3T^{2} \)
79 \( 1 + (24.6 - 42.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-14.3 + 24.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 65.9iT - 7.92e3T^{2} \)
97 \( 1 + (-62.1 - 35.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.07720227047396914285415708969, −10.37289119444379628485602962427, −9.449341405193347310605436584022, −7.916399132630995179875997652755, −7.35131332810875175825910172951, −6.09129837951561311877089656926, −4.69300490959505680760664571593, −3.45386007166000480830672443843, −3.12459041476811095110657837123, −1.23702057088069908978709091197, 1.14399017759859821158476892644, 3.38775323101041808693847287669, 4.46306797737699084490240493779, 5.43085733963537200845270467002, 6.35035731965239806250801458697, 7.27811989301880270108361772336, 8.224465638247794074692085675165, 9.028035279566102019460542314189, 10.23825122217749940619218291134, 11.28287417573470978511102905360

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