Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
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 + 0.295·3-s + 4-s + 0.172·5-s + 0.295·6-s − 3.30·7-s + 8-s − 2.91·9-s + 0.172·10-s − 2.45·11-s + 0.295·12-s − 2.40·13-s − 3.30·14-s + 0.0508·15-s + 16-s − 0.737·17-s − 2.91·18-s − 19-s + 0.172·20-s − 0.976·21-s − 2.45·22-s − 1.13·23-s + 0.295·24-s − 4.97·25-s − 2.40·26-s − 1.74·27-s − 3.30·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.170·3-s + 0.5·4-s + 0.0769·5-s + 0.120·6-s − 1.24·7-s + 0.353·8-s − 0.970·9-s + 0.0544·10-s − 0.739·11-s + 0.0852·12-s − 0.666·13-s − 0.883·14-s + 0.0131·15-s + 0.250·16-s − 0.178·17-s − 0.686·18-s − 0.229·19-s + 0.0384·20-s − 0.213·21-s − 0.522·22-s − 0.235·23-s + 0.0602·24-s − 0.994·25-s − 0.470·26-s − 0.336·27-s − 0.624·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.857541983$
$L(\frac12)$  $\approx$  $1.857541983$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 - 0.295T + 3T^{2} \)
5 \( 1 - 0.172T + 5T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 + 2.45T + 11T^{2} \)
13 \( 1 + 2.40T + 13T^{2} \)
17 \( 1 + 0.737T + 17T^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 - 2.23T + 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
41 \( 1 - 8.47T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 2.36T + 59T^{2} \)
61 \( 1 - 7.18T + 61T^{2} \)
67 \( 1 + 0.726T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 3.97T + 73T^{2} \)
79 \( 1 - 5.35T + 79T^{2} \)
83 \( 1 - 2.06T + 83T^{2} \)
89 \( 1 - 3.34T + 89T^{2} \)
97 \( 1 - 6.07T + 97T^{2} \)
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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

−7.78042895366794439483859568467, −7.02141164990803730823723014621, −6.26557663940010490547380473129, −5.81459830672674591176555901372, −5.11074910078587835652975418899, −4.21139959215292499673210007358, −3.47859400849658489123192980784, −2.62707098275496585000647060655, −2.33166873302483101360351888446, −0.55544953862441945080119166967, 0.55544953862441945080119166967, 2.33166873302483101360351888446, 2.62707098275496585000647060655, 3.47859400849658489123192980784, 4.21139959215292499673210007358, 5.11074910078587835652975418899, 5.81459830672674591176555901372, 6.26557663940010490547380473129, 7.02141164990803730823723014621, 7.78042895366794439483859568467

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