Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.744·3-s + 4-s + 3.33·5-s − 0.744·6-s + 0.241·7-s − 8-s − 2.44·9-s − 3.33·10-s − 2.45·11-s + 0.744·12-s + 5.12·13-s − 0.241·14-s + 2.48·15-s + 16-s − 1.97·17-s + 2.44·18-s − 19-s + 3.33·20-s + 0.179·21-s + 2.45·22-s − 8.65·23-s − 0.744·24-s + 6.10·25-s − 5.12·26-s − 4.05·27-s + 0.241·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.429·3-s + 0.5·4-s + 1.49·5-s − 0.303·6-s + 0.0913·7-s − 0.353·8-s − 0.815·9-s − 1.05·10-s − 0.741·11-s + 0.214·12-s + 1.42·13-s − 0.0646·14-s + 0.640·15-s + 0.250·16-s − 0.477·17-s + 0.576·18-s − 0.229·19-s + 0.745·20-s + 0.0392·21-s + 0.524·22-s − 1.80·23-s − 0.151·24-s + 1.22·25-s − 1.00·26-s − 0.780·27-s + 0.0456·28-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  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 - 0.744T + 3T^{2} \)
5 \( 1 - 3.33T + 5T^{2} \)
7 \( 1 - 0.241T + 7T^{2} \)
11 \( 1 + 2.45T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 1.97T + 17T^{2} \)
23 \( 1 + 8.65T + 23T^{2} \)
29 \( 1 - 0.510T + 29T^{2} \)
31 \( 1 - 2.52T + 31T^{2} \)
37 \( 1 + 1.19T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 + 6.78T + 43T^{2} \)
47 \( 1 + 1.86T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 + 8.01T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 2.48T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 1.80T + 97T^{2} \)
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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

−7.79913122102042577681578133989, −6.62483142882256922993986294388, −6.13413815108083989356016631493, −5.71117599383424257837444827345, −4.82205414716555308789186578251, −3.64210914827374804435820954046, −2.84073325238221661061599220022, −2.07384431611017412664549987234, −1.50554334557011209883953803228, 0, 1.50554334557011209883953803228, 2.07384431611017412664549987234, 2.84073325238221661061599220022, 3.64210914827374804435820954046, 4.82205414716555308789186578251, 5.71117599383424257837444827345, 6.13413815108083989356016631493, 6.62483142882256922993986294388, 7.79913122102042577681578133989

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