Properties

Degree 2
Conductor $ 7 \cdot 863 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·2-s + 1.90·3-s + 3.00·4-s + 4.23·5-s − 4.26·6-s + 7-s − 2.25·8-s + 0.626·9-s − 9.47·10-s − 0.780·11-s + 5.72·12-s − 5.45·13-s − 2.23·14-s + 8.06·15-s − 0.966·16-s + 1.79·17-s − 1.40·18-s − 4.44·19-s + 12.7·20-s + 1.90·21-s + 1.74·22-s + 7.54·23-s − 4.29·24-s + 12.9·25-s + 12.1·26-s − 4.52·27-s + 3.00·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.09·3-s + 1.50·4-s + 1.89·5-s − 1.73·6-s + 0.377·7-s − 0.797·8-s + 0.208·9-s − 2.99·10-s − 0.235·11-s + 1.65·12-s − 1.51·13-s − 0.598·14-s + 2.08·15-s − 0.241·16-s + 0.436·17-s − 0.330·18-s − 1.02·19-s + 2.84·20-s + 0.415·21-s + 0.372·22-s + 1.57·23-s − 0.877·24-s + 2.58·25-s + 2.39·26-s − 0.869·27-s + 0.568·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6041\)    =    \(7 \cdot 863\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6041,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.016795120$
$L(\frac12)$  $\approx$  $2.016795120$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;863\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;863\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
863 \( 1 + T \)
good2 \( 1 + 2.23T + 2T^{2} \)
3 \( 1 - 1.90T + 3T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
11 \( 1 + 0.780T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 - 1.79T + 17T^{2} \)
19 \( 1 + 4.44T + 19T^{2} \)
23 \( 1 - 7.54T + 23T^{2} \)
29 \( 1 - 4.65T + 29T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 + 2.02T + 37T^{2} \)
41 \( 1 - 9.44T + 41T^{2} \)
43 \( 1 + 8.07T + 43T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 8.64T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 + 4.69T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 8.77T + 73T^{2} \)
79 \( 1 + 6.49T + 79T^{2} \)
83 \( 1 + 0.00279T + 83T^{2} \)
89 \( 1 - 8.85T + 89T^{2} \)
97 \( 1 - 5.10T + 97T^{2} \)
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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

−8.312329474999551595673487759714, −7.65801995670187326443065765508, −6.86227804251204009352617244483, −6.30130706124821000490178811310, −5.23808896464111024795911457819, −4.62410316426306931379761494655, −2.86642903015480479211223456228, −2.52299764248025546819610844780, −1.88251986395840615263202689883, −0.914118775405580418287630417406, 0.914118775405580418287630417406, 1.88251986395840615263202689883, 2.52299764248025546819610844780, 2.86642903015480479211223456228, 4.62410316426306931379761494655, 5.23808896464111024795911457819, 6.30130706124821000490178811310, 6.86227804251204009352617244483, 7.65801995670187326443065765508, 8.312329474999551595673487759714

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