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Properties

Label 2-405-5.2-c2-0-23
Degree $2$
Conductor $405$
Sign $0.956 - 0.292i$
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  + (0.652 + 0.652i)2-s − 3.14i·4-s + (4.83 + 1.27i)5-s + (8.09 + 8.09i)7-s + (4.66 − 4.66i)8-s + (2.32 + 3.98i)10-s + 2.60·11-s + (−2.08 + 2.08i)13-s + 10.5i·14-s − 6.49·16-s + (−13.6 − 13.6i)17-s + 11.4i·19-s + (3.99 − 15.2i)20-s + (1.70 + 1.70i)22-s + (7.89 − 7.89i)23-s + ⋯
L(s)  = 1  + (0.326 + 0.326i)2-s − 0.786i·4-s + (0.967 + 0.254i)5-s + (1.15 + 1.15i)7-s + (0.583 − 0.583i)8-s + (0.232 + 0.398i)10-s + 0.237·11-s + (−0.160 + 0.160i)13-s + 0.754i·14-s − 0.406·16-s + (−0.805 − 0.805i)17-s + 0.602i·19-s + (0.199 − 0.761i)20-s + (0.0774 + 0.0774i)22-s + (0.343 − 0.343i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.69952 + 0.403375i\)
\(L(\frac12)\) \(\approx\) \(2.69952 + 0.403375i\)
\(L(2)\) 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 \( 1 + (-4.83 - 1.27i)T \)
good2 \( 1 + (-0.652 - 0.652i)T + 4iT^{2} \)
7 \( 1 + (-8.09 - 8.09i)T + 49iT^{2} \)
11 \( 1 - 2.60T + 121T^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 169iT^{2} \)
17 \( 1 + (13.6 + 13.6i)T + 289iT^{2} \)
19 \( 1 - 11.4iT - 361T^{2} \)
23 \( 1 + (-7.89 + 7.89i)T - 529iT^{2} \)
29 \( 1 - 26.5iT - 841T^{2} \)
31 \( 1 - 43.7T + 961T^{2} \)
37 \( 1 + (-14.4 - 14.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 1.84T + 1.68e3T^{2} \)
43 \( 1 + (-35.9 + 35.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.8 + 44.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-6.43 + 6.43i)T - 2.80e3iT^{2} \)
59 \( 1 + 57.1iT - 3.48e3T^{2} \)
61 \( 1 + 33.6T + 3.72e3T^{2} \)
67 \( 1 + (22.9 + 22.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 63.3T + 5.04e3T^{2} \)
73 \( 1 + (50.9 - 50.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 68.8iT - 6.24e3T^{2} \)
83 \( 1 + (74.5 - 74.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 136. iT - 7.92e3T^{2} \)
97 \( 1 + (25.8 + 25.8i)T + 9.40e3iT^{2} \)
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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

−11.07359792095811696697057798729, −10.15631564041716811355084945315, −9.267681389383253781833083401928, −8.490452630708618572840013640246, −7.04553888659180798430099706525, −6.17116816303893349813351769856, −5.30661080729638748473482800843, −4.64239197881359452841625666838, −2.53338601889853475737421206577, −1.50644303311655297926675010220, 1.37599633209726248094207377649, 2.64460653632616627193093762664, 4.24013771183338890987560134349, 4.75479292894106357415203751104, 6.22499167678456328697567945600, 7.38988556705260103668852352396, 8.182286731067651278862891341576, 9.126196881316583498106406675689, 10.33363109731263497294925215571, 11.04751253085095456844759851475

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