Properties

Label 2-4018-1.1-c1-0-120
Degree $2$
Conductor $4018$
Sign $-1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.53·3-s + 4-s − 1.21·5-s − 2.53·6-s − 8-s + 3.42·9-s + 1.21·10-s + 3.74·11-s + 2.53·12-s − 2.95·13-s − 3.06·15-s + 16-s − 6.53·17-s − 3.42·18-s − 4.95·19-s − 1.21·20-s − 3.74·22-s + 6.02·23-s − 2.53·24-s − 3.53·25-s + 2.95·26-s + 1.06·27-s − 0.901·29-s + 3.06·30-s − 9.48·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.46·3-s + 0.5·4-s − 0.541·5-s − 1.03·6-s − 0.353·8-s + 1.14·9-s + 0.382·10-s + 1.12·11-s + 0.731·12-s − 0.819·13-s − 0.792·15-s + 0.250·16-s − 1.58·17-s − 0.806·18-s − 1.13·19-s − 0.270·20-s − 0.798·22-s + 1.25·23-s − 0.517·24-s − 0.706·25-s + 0.579·26-s + 0.205·27-s − 0.167·29-s + 0.560·30-s − 1.70·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(32.0838\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 2.53T + 3T^{2} \)
5 \( 1 + 1.21T + 5T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 + 6.53T + 17T^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 - 6.02T + 23T^{2} \)
29 \( 1 + 0.901T + 29T^{2} \)
31 \( 1 + 9.48T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
43 \( 1 + 9.06T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 5.32T + 53T^{2} \)
59 \( 1 + 2.36T + 59T^{2} \)
61 \( 1 + 3.76T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 1.06T + 73T^{2} \)
79 \( 1 - 4.84T + 79T^{2} \)
83 \( 1 - 1.29T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 4.13T + 97T^{2} \)
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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

−8.241897343057805096643253183059, −7.53619100298152590604165517764, −6.90150674072391108696303836394, −6.25203328286861974565347052523, −4.78472896555252136205883272798, −4.04339558848267215510382112909, −3.29180279926585272497948266651, −2.34459212189848846270867947138, −1.67824977931410804250975117303, 0, 1.67824977931410804250975117303, 2.34459212189848846270867947138, 3.29180279926585272497948266651, 4.04339558848267215510382112909, 4.78472896555252136205883272798, 6.25203328286861974565347052523, 6.90150674072391108696303836394, 7.53619100298152590604165517764, 8.241897343057805096643253183059

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