Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s + 5.85·5-s − 6·6-s + 28.5·7-s − 8·8-s + 9·9-s − 11.7·10-s − 2.56·11-s + 12·12-s + 82.9·13-s − 57.1·14-s + 17.5·15-s + 16·16-s − 74.8·17-s − 18·18-s + 34·19-s + 23.4·20-s + 85.6·21-s + 5.13·22-s + 15.1·23-s − 24·24-s − 90.6·25-s − 165.·26-s + 27·27-s + 114.·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.524·5-s − 0.408·6-s + 1.54·7-s − 0.353·8-s + 0.333·9-s − 0.370·10-s − 0.0703·11-s + 0.288·12-s + 1.77·13-s − 1.09·14-s + 0.302·15-s + 0.250·16-s − 1.06·17-s − 0.235·18-s + 0.410·19-s + 0.262·20-s + 0.890·21-s + 0.0497·22-s + 0.137·23-s − 0.204·24-s − 0.725·25-s − 1.25·26-s + 0.192·27-s + 0.771·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{354} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $2.322692999$
$L(\frac12)$  $\approx$  $2.322692999$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 - 3T \)
59 \( 1 - 59T \)
good5 \( 1 - 5.85T + 125T^{2} \)
7 \( 1 - 28.5T + 343T^{2} \)
11 \( 1 + 2.56T + 1.33e3T^{2} \)
13 \( 1 - 82.9T + 2.19e3T^{2} \)
17 \( 1 + 74.8T + 4.91e3T^{2} \)
19 \( 1 - 34T + 6.85e3T^{2} \)
23 \( 1 - 15.1T + 1.21e4T^{2} \)
29 \( 1 - 239.T + 2.43e4T^{2} \)
31 \( 1 + 242.T + 2.97e4T^{2} \)
37 \( 1 + 269.T + 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
43 \( 1 - 441.T + 7.95e4T^{2} \)
47 \( 1 + 249.T + 1.03e5T^{2} \)
53 \( 1 + 19.2T + 1.48e5T^{2} \)
61 \( 1 - 789.T + 2.26e5T^{2} \)
67 \( 1 - 829.T + 3.00e5T^{2} \)
71 \( 1 + 825.T + 3.57e5T^{2} \)
73 \( 1 - 913.T + 3.89e5T^{2} \)
79 \( 1 + 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 873.T + 5.71e5T^{2} \)
89 \( 1 + 888.T + 7.04e5T^{2} \)
97 \( 1 - 384.T + 9.12e5T^{2} \)
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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

−11.00478231461511973422692794527, −10.11200340950897202232202702423, −8.824779134648160846813919739788, −8.537506757522985625002198102990, −7.52226873176453074277556292778, −6.37215331837829028975083721783, −5.16693076999975991205616861697, −3.80126073757737553701627632781, −2.17445645535342592666006834043, −1.26708412265135184728524806385, 1.26708412265135184728524806385, 2.17445645535342592666006834043, 3.80126073757737553701627632781, 5.16693076999975991205616861697, 6.37215331837829028975083721783, 7.52226873176453074277556292778, 8.537506757522985625002198102990, 8.824779134648160846813919739788, 10.11200340950897202232202702423, 11.00478231461511973422692794527

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