Properties

Label 2-4026-1.1-c1-0-18
Degree $2$
Conductor $4026$
Sign $1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 3.56·5-s − 6-s + 3.12·7-s + 8-s + 9-s − 3.56·10-s + 11-s − 12-s + 0.438·13-s + 3.12·14-s + 3.56·15-s + 16-s + 5.12·17-s + 18-s − 4·19-s − 3.56·20-s − 3.12·21-s + 22-s − 24-s + 7.68·25-s + 0.438·26-s − 27-s + 3.12·28-s + 0.438·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.59·5-s − 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s − 1.12·10-s + 0.301·11-s − 0.288·12-s + 0.121·13-s + 0.834·14-s + 0.919·15-s + 0.250·16-s + 1.24·17-s + 0.235·18-s − 0.917·19-s − 0.796·20-s − 0.681·21-s + 0.213·22-s − 0.204·24-s + 1.53·25-s + 0.0859·26-s − 0.192·27-s + 0.590·28-s + 0.0814·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.108468909\)
\(L(\frac12)\) \(\approx\) \(2.108468909\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
61 \( 1 - T \)
good5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 0.438T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 - 1.56T + 59T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 3.12T + 71T^{2} \)
73 \( 1 - 5.12T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 6.68T + 97T^{2} \)
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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

−8.186271738488024773627791442517, −7.61352489752610147560192472069, −7.09878355214631797302430913557, −6.11124140920893868339343789696, −5.29442640203350541182991335888, −4.61063074669619040841600316742, −3.99439497616293940872971858151, −3.32799247573596868477010843854, −1.94583125093181597011122366542, −0.789791873803405629986404740523, 0.789791873803405629986404740523, 1.94583125093181597011122366542, 3.32799247573596868477010843854, 3.99439497616293940872971858151, 4.61063074669619040841600316742, 5.29442640203350541182991335888, 6.11124140920893868339343789696, 7.09878355214631797302430913557, 7.61352489752610147560192472069, 8.186271738488024773627791442517

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