Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2.30·5-s − 6-s + 4.49·7-s − 8-s + 9-s − 2.30·10-s − 3.07·11-s + 12-s + 13-s − 4.49·14-s + 2.30·15-s + 16-s − 6.42·17-s − 18-s − 7.58·19-s + 2.30·20-s + 4.49·21-s + 3.07·22-s − 9.27·23-s − 24-s + 0.326·25-s − 26-s + 27-s + 4.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.03·5-s − 0.408·6-s + 1.70·7-s − 0.353·8-s + 0.333·9-s − 0.729·10-s − 0.925·11-s + 0.288·12-s + 0.277·13-s − 1.20·14-s + 0.595·15-s + 0.250·16-s − 1.55·17-s − 0.235·18-s − 1.73·19-s + 0.516·20-s + 0.981·21-s + 0.654·22-s − 1.93·23-s − 0.204·24-s + 0.0653·25-s − 0.196·26-s + 0.192·27-s + 0.850·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 - 4.49T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
17 \( 1 + 6.42T + 17T^{2} \)
19 \( 1 + 7.58T + 19T^{2} \)
23 \( 1 + 9.27T + 23T^{2} \)
29 \( 1 + 3.29T + 29T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 - 2.93T + 37T^{2} \)
41 \( 1 - 7.89T + 41T^{2} \)
43 \( 1 + 0.660T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 3.71T + 53T^{2} \)
59 \( 1 + 8.12T + 59T^{2} \)
61 \( 1 - 4.61T + 61T^{2} \)
67 \( 1 + 6.12T + 67T^{2} \)
71 \( 1 + 8.35T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 1.15T + 89T^{2} \)
97 \( 1 - 5.29T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.87619966872792659612998549964, −6.92760205787768564217446702297, −6.09488363410652045599559198350, −5.56786348323387040565894313215, −4.54963002300662403622672199142, −4.07176991409974535847177776325, −2.57855226391466602717450463828, −1.95233650972284143874528992049, −1.74988949318623385051823895131, 0, 1.74988949318623385051823895131, 1.95233650972284143874528992049, 2.57855226391466602717450463828, 4.07176991409974535847177776325, 4.54963002300662403622672199142, 5.56786348323387040565894313215, 6.09488363410652045599559198350, 6.92760205787768564217446702297, 7.87619966872792659612998549964

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