Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.390·2-s − 3-s − 1.84·4-s + 3.91·5-s − 0.390·6-s + 2.71·7-s − 1.50·8-s + 9-s + 1.53·10-s − 5.03·11-s + 1.84·12-s − 2.69·13-s + 1.06·14-s − 3.91·15-s + 3.10·16-s − 17-s + 0.390·18-s − 4.89·19-s − 7.22·20-s − 2.71·21-s − 1.96·22-s + 3.69·23-s + 1.50·24-s + 10.3·25-s − 1.05·26-s − 27-s − 5.01·28-s + ⋯
L(s)  = 1  + 0.276·2-s − 0.577·3-s − 0.923·4-s + 1.75·5-s − 0.159·6-s + 1.02·7-s − 0.531·8-s + 0.333·9-s + 0.483·10-s − 1.51·11-s + 0.533·12-s − 0.746·13-s + 0.283·14-s − 1.01·15-s + 0.776·16-s − 0.242·17-s + 0.0921·18-s − 1.12·19-s − 1.61·20-s − 0.592·21-s − 0.419·22-s + 0.771·23-s + 0.307·24-s + 2.06·25-s − 0.206·26-s − 0.192·27-s − 0.947·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 0.390T + 2T^{2} \)
5 \( 1 - 3.91T + 5T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 5.03T + 11T^{2} \)
13 \( 1 + 2.69T + 13T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 - 3.69T + 23T^{2} \)
29 \( 1 - 0.627T + 29T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + 4.11T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 + 0.926T + 53T^{2} \)
59 \( 1 - 0.268T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 4.07T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−7.47543336273212983662595639679, −6.57641658731494388150612628932, −5.86264222401462742581629000191, −5.19576249248243470674791556180, −4.99440434327350919403849849642, −4.30934500994911653755654232813, −2.89107971468542239641994473475, −2.23042021298111678123180388562, −1.31851377013766485224896155454, 0, 1.31851377013766485224896155454, 2.23042021298111678123180388562, 2.89107971468542239641994473475, 4.30934500994911653755654232813, 4.99440434327350919403849849642, 5.19576249248243470674791556180, 5.86264222401462742581629000191, 6.57641658731494388150612628932, 7.47543336273212983662595639679

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