Properties

Degree 2
Conductor $ 19 \cdot 317 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95·2-s + 1.09·3-s + 1.83·4-s + 1.35·5-s − 2.14·6-s + 3.30·7-s + 0.315·8-s − 1.80·9-s − 2.64·10-s − 1.41·11-s + 2.00·12-s − 1.08·13-s − 6.47·14-s + 1.47·15-s − 4.29·16-s + 0.206·17-s + 3.53·18-s − 19-s + 2.48·20-s + 3.61·21-s + 2.77·22-s − 2.29·23-s + 0.345·24-s − 3.17·25-s + 2.12·26-s − 5.25·27-s + 6.07·28-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.630·3-s + 0.919·4-s + 0.603·5-s − 0.873·6-s + 1.24·7-s + 0.111·8-s − 0.602·9-s − 0.836·10-s − 0.427·11-s + 0.579·12-s − 0.301·13-s − 1.73·14-s + 0.380·15-s − 1.07·16-s + 0.0501·17-s + 0.834·18-s − 0.229·19-s + 0.555·20-s + 0.788·21-s + 0.591·22-s − 0.478·23-s + 0.0704·24-s − 0.635·25-s + 0.417·26-s − 1.01·27-s + 1.14·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6023\)    =    \(19 \cdot 317\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;317\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;317\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 + T \)
317 \( 1 + T \)
good2 \( 1 + 1.95T + 2T^{2} \)
3 \( 1 - 1.09T + 3T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 1.08T + 13T^{2} \)
17 \( 1 - 0.206T + 17T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 - 3.93T + 29T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 + 2.28T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 8.76T + 43T^{2} \)
47 \( 1 + 0.177T + 47T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 + 8.86T + 59T^{2} \)
61 \( 1 + 1.63T + 61T^{2} \)
67 \( 1 - 3.90T + 67T^{2} \)
71 \( 1 + 5.36T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 7.59T + 79T^{2} \)
83 \( 1 - 5.04T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 0.326T + 97T^{2} \)
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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

−7.901269879313547135108925844651, −7.51204713437975862153073669934, −6.52424249926737965513848242723, −5.60945981381380206740354922709, −4.92128519763135600483750788266, −4.01750561867098550324556310837, −2.74678457705027645992663970732, −2.08001268190915350776271802969, −1.39794397877292747660067452595, 0, 1.39794397877292747660067452595, 2.08001268190915350776271802969, 2.74678457705027645992663970732, 4.01750561867098550324556310837, 4.92128519763135600483750788266, 5.60945981381380206740354922709, 6.52424249926737965513848242723, 7.51204713437975862153073669934, 7.901269879313547135108925844651

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