Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
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 − 4.06·5-s − 6-s + 4.31·7-s + 8-s + 9-s − 4.06·10-s − 3.45·11-s − 12-s − 13-s + 4.31·14-s + 4.06·15-s + 16-s − 7.56·17-s + 18-s − 1.84·19-s − 4.06·20-s − 4.31·21-s − 3.45·22-s − 8.50·23-s − 24-s + 11.5·25-s − 26-s − 27-s + 4.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.81·5-s − 0.408·6-s + 1.63·7-s + 0.353·8-s + 0.333·9-s − 1.28·10-s − 1.04·11-s − 0.288·12-s − 0.277·13-s + 1.15·14-s + 1.05·15-s + 0.250·16-s − 1.83·17-s + 0.235·18-s − 0.424·19-s − 0.909·20-s − 0.942·21-s − 0.737·22-s − 1.77·23-s − 0.204·24-s + 2.31·25-s − 0.196·26-s − 0.192·27-s + 0.816·28-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  =  \(0\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.317558521\)
\(L(\frac12)\)  \(\approx\)  \(1.317558521\)
\(L(\frac{3}{2})\)   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 \( 1 + 4.06T + 5T^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
17 \( 1 + 7.56T + 17T^{2} \)
19 \( 1 + 1.84T + 19T^{2} \)
23 \( 1 + 8.50T + 23T^{2} \)
29 \( 1 - 6.61T + 29T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 0.695T + 41T^{2} \)
43 \( 1 + 3.09T + 43T^{2} \)
47 \( 1 + 1.17T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 - 6.88T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 - 4.40T + 71T^{2} \)
73 \( 1 + 8.29T + 73T^{2} \)
79 \( 1 - 6.21T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 2.88T + 89T^{2} \)
97 \( 1 - 4.85T + 97T^{2} \)
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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

−7.76872753740719835976623666381, −7.23828203036481850885397743566, −6.46751677833063377020521940435, −5.52445448314367404135248009753, −4.78556584833424349318372158404, −4.35336632450483268336468538770, −3.98049302389213811035828656360, −2.67727364094744646549525183990, −1.92852141511718703416718700533, −0.50635352261230398466069064662, 0.50635352261230398466069064662, 1.92852141511718703416718700533, 2.67727364094744646549525183990, 3.98049302389213811035828656360, 4.35336632450483268336468538770, 4.78556584833424349318372158404, 5.52445448314367404135248009753, 6.46751677833063377020521940435, 7.23828203036481850885397743566, 7.76872753740719835976623666381

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