Properties

Label 2-4014-1.1-c1-0-35
Degree $2$
Conductor $4014$
Sign $-1$
Analytic cond. $32.0519$
Root an. cond. $5.66144$
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 + 4-s − 3.85·5-s − 2.71·7-s − 8-s + 3.85·10-s + 0.310·11-s + 2.17·13-s + 2.71·14-s + 16-s + 0.173·17-s − 1.74·19-s − 3.85·20-s − 0.310·22-s − 0.551·23-s + 9.85·25-s − 2.17·26-s − 2.71·28-s + 1.97·29-s + 5.74·31-s − 32-s − 0.173·34-s + 10.4·35-s + 3.77·37-s + 1.74·38-s + 3.85·40-s − 0.377·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.72·5-s − 1.02·7-s − 0.353·8-s + 1.21·10-s + 0.0936·11-s + 0.602·13-s + 0.725·14-s + 0.250·16-s + 0.0421·17-s − 0.399·19-s − 0.861·20-s − 0.0662·22-s − 0.115·23-s + 1.97·25-s − 0.425·26-s − 0.513·28-s + 0.366·29-s + 1.03·31-s − 0.176·32-s − 0.0297·34-s + 1.76·35-s + 0.620·37-s + 0.282·38-s + 0.609·40-s − 0.0590·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
Sign: $-1$
Analytic conductor: \(32.0519\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4014,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
223 \( 1 - T \)
good5 \( 1 + 3.85T + 5T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
11 \( 1 - 0.310T + 11T^{2} \)
13 \( 1 - 2.17T + 13T^{2} \)
17 \( 1 - 0.173T + 17T^{2} \)
19 \( 1 + 1.74T + 19T^{2} \)
23 \( 1 + 0.551T + 23T^{2} \)
29 \( 1 - 1.97T + 29T^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
37 \( 1 - 3.77T + 37T^{2} \)
41 \( 1 + 0.377T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 - 2.59T + 47T^{2} \)
53 \( 1 - 4.08T + 53T^{2} \)
59 \( 1 - 1.87T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 1.97T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + 4.88T + 97T^{2} \)
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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

−8.108066362221360855457564926934, −7.53922075534008444366876800523, −6.64629083156053741068168182050, −6.29001898753218656891850678338, −4.97925577212787589731440915062, −4.00188242574905114509137251438, −3.45367358356901074488987639488, −2.59130729687721254575211245552, −0.999145651603834286080813506855, 0, 0.999145651603834286080813506855, 2.59130729687721254575211245552, 3.45367358356901074488987639488, 4.00188242574905114509137251438, 4.97925577212787589731440915062, 6.29001898753218656891850678338, 6.64629083156053741068168182050, 7.53922075534008444366876800523, 8.108066362221360855457564926934

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