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 + 20.1·5-s − 6·6-s − 28.5·7-s − 8·8-s + 9·9-s − 40.2·10-s + 54.5·11-s + 12·12-s − 16.9·13-s + 57.1·14-s + 60.4·15-s + 16·16-s + 10.8·17-s − 18·18-s + 34·19-s + 80.5·20-s − 85.6·21-s − 109.·22-s + 100.·23-s − 24·24-s + 280.·25-s + 33.9·26-s + 27·27-s − 114.·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.80·5-s − 0.408·6-s − 1.54·7-s − 0.353·8-s + 0.333·9-s − 1.27·10-s + 1.49·11-s + 0.288·12-s − 0.362·13-s + 1.09·14-s + 1.04·15-s + 0.250·16-s + 0.154·17-s − 0.235·18-s + 0.410·19-s + 0.900·20-s − 0.890·21-s − 1.05·22-s + 0.914·23-s − 0.204·24-s + 2.24·25-s + 0.256·26-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  =  $(2,\ 354,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $2.176732221$
$L(\frac12)$  $\approx$  $2.176732221$
$L(\frac{5}{2})$   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 \( 1 - 20.1T + 125T^{2} \)
7 \( 1 + 28.5T + 343T^{2} \)
11 \( 1 - 54.5T + 1.33e3T^{2} \)
13 \( 1 + 16.9T + 2.19e3T^{2} \)
17 \( 1 - 10.8T + 4.91e3T^{2} \)
19 \( 1 - 34T + 6.85e3T^{2} \)
23 \( 1 - 100.T + 1.21e4T^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 - 0.404T + 2.97e4T^{2} \)
37 \( 1 + 368.T + 5.06e4T^{2} \)
41 \( 1 + 436.T + 6.89e4T^{2} \)
43 \( 1 - 326.T + 7.95e4T^{2} \)
47 \( 1 - 293.T + 1.03e5T^{2} \)
53 \( 1 - 109.T + 1.48e5T^{2} \)
61 \( 1 - 460.T + 2.26e5T^{2} \)
67 \( 1 - 630.T + 3.00e5T^{2} \)
71 \( 1 - 987.T + 3.57e5T^{2} \)
73 \( 1 + 629.T + 3.89e5T^{2} \)
79 \( 1 - 715.T + 4.93e5T^{2} \)
83 \( 1 + 497.T + 5.71e5T^{2} \)
89 \( 1 - 396.T + 7.04e5T^{2} \)
97 \( 1 + 1.47e3T + 9.12e5T^{2} \)
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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

−10.52877254078328178390087423128, −9.817152716644443673319396659264, −9.322543066975650477180345815801, −8.735091791473356010379578870730, −6.88216624453555474779064443854, −6.60395135955651265288225434024, −5.43326695718227521189907725024, −3.48927258245031158604539977496, −2.42275211842072018208899188403, −1.15937265681046404676898343652, 1.15937265681046404676898343652, 2.42275211842072018208899188403, 3.48927258245031158604539977496, 5.43326695718227521189907725024, 6.60395135955651265288225434024, 6.88216624453555474779064443854, 8.735091791473356010379578870730, 9.322543066975650477180345815801, 9.817152716644443673319396659264, 10.52877254078328178390087423128

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