Properties

Label 2-4018-1.1-c1-0-65
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 + 0.539·3-s + 4-s − 3.38·5-s − 0.539·6-s − 8-s − 2.70·9-s + 3.38·10-s + 2.98·11-s + 0.539·12-s + 0.00964·13-s − 1.82·15-s + 16-s + 0.0684·17-s + 2.70·18-s − 0.966·19-s − 3.38·20-s − 2.98·22-s + 1.61·23-s − 0.539·24-s + 6.46·25-s − 0.00964·26-s − 3.08·27-s + 0.701·29-s + 1.82·30-s + 7.29·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.311·3-s + 0.5·4-s − 1.51·5-s − 0.220·6-s − 0.353·8-s − 0.902·9-s + 1.07·10-s + 0.900·11-s + 0.155·12-s + 0.00267·13-s − 0.471·15-s + 0.250·16-s + 0.0165·17-s + 0.638·18-s − 0.221·19-s − 0.757·20-s − 0.636·22-s + 0.337·23-s − 0.110·24-s + 1.29·25-s − 0.00189·26-s − 0.592·27-s + 0.130·29-s + 0.333·30-s + 1.30·31-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 \( 1 - 0.539T + 3T^{2} \)
5 \( 1 + 3.38T + 5T^{2} \)
11 \( 1 - 2.98T + 11T^{2} \)
13 \( 1 - 0.00964T + 13T^{2} \)
17 \( 1 - 0.0684T + 17T^{2} \)
19 \( 1 + 0.966T + 19T^{2} \)
23 \( 1 - 1.61T + 23T^{2} \)
29 \( 1 - 0.701T + 29T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 - 4.88T + 37T^{2} \)
43 \( 1 - 1.84T + 43T^{2} \)
47 \( 1 + 3.16T + 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 + 3.16T + 59T^{2} \)
61 \( 1 + 8.67T + 61T^{2} \)
67 \( 1 - 8.55T + 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 6.62T + 83T^{2} \)
89 \( 1 + 9.28T + 89T^{2} \)
97 \( 1 - 4.16T + 97T^{2} \)
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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

−8.009026681524729415501153903709, −7.72694051283163490708573172480, −6.71418070496032263424939582574, −6.17597611355614720845338517663, −4.95379274311440526932133971821, −4.08605742837137205019351567823, −3.34285543096341277714693329730, −2.56696916703774885288934006237, −1.14940014008191375502468423631, 0, 1.14940014008191375502468423631, 2.56696916703774885288934006237, 3.34285543096341277714693329730, 4.08605742837137205019351567823, 4.95379274311440526932133971821, 6.17597611355614720845338517663, 6.71418070496032263424939582574, 7.72694051283163490708573172480, 8.009026681524729415501153903709

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