Properties

Degree 2
Conductor 3
Sign $-1$
Motivic weight 19
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.10e3·2-s + 1.96e4·3-s + 6.94e5·4-s + 3.51e6·5-s − 2.17e7·6-s − 1.95e8·7-s − 1.87e8·8-s + 3.87e8·9-s − 3.88e9·10-s − 2.74e9·11-s + 1.36e10·12-s − 4.44e10·13-s + 2.15e11·14-s + 6.92e10·15-s − 1.56e11·16-s − 7.85e11·17-s − 4.27e11·18-s + 3.15e11·19-s + 2.44e12·20-s − 3.84e12·21-s + 3.03e12·22-s + 4.90e12·23-s − 3.69e12·24-s − 6.70e12·25-s + 4.90e13·26-s + 7.62e12·27-s − 1.35e14·28-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.577·3-s + 1.32·4-s + 0.805·5-s − 0.880·6-s − 1.83·7-s − 0.495·8-s + 1/3·9-s − 1.22·10-s − 0.351·11-s + 0.764·12-s − 1.16·13-s + 2.79·14-s + 0.464·15-s − 0.569·16-s − 1.60·17-s − 0.508·18-s + 0.224·19-s + 1.06·20-s − 1.05·21-s + 0.535·22-s + 0.567·23-s − 0.285·24-s − 0.351·25-s + 1.77·26-s + 0.192·27-s − 2.42·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(19\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3,\ (\ :19/2),\ -1)$
$L(10)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{21}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - p^{9} T \)
good2 \( 1 + 69 p^{4} T + p^{19} T^{2} \)
5 \( 1 - 703254 p T + p^{19} T^{2} \)
7 \( 1 + 27941512 p T + p^{19} T^{2} \)
11 \( 1 + 2746857948 T + p^{19} T^{2} \)
13 \( 1 + 3415418866 p T + p^{19} T^{2} \)
17 \( 1 + 46234265742 p T + p^{19} T^{2} \)
19 \( 1 - 315410465180 T + p^{19} T^{2} \)
23 \( 1 - 4900560535752 T + p^{19} T^{2} \)
29 \( 1 - 12188520672150 T + p^{19} T^{2} \)
31 \( 1 + 42713658601168 T + p^{19} T^{2} \)
37 \( 1 + 423452395388194 T + p^{19} T^{2} \)
41 \( 1 + 1113920690896038 T + p^{19} T^{2} \)
43 \( 1 - 1136100238138052 T + p^{19} T^{2} \)
47 \( 1 - 1531372040448816 T + p^{19} T^{2} \)
53 \( 1 + 18059320314853218 T + p^{19} T^{2} \)
59 \( 1 - 92700438637662420 T + p^{19} T^{2} \)
61 \( 1 - 21352962331944422 T + p^{19} T^{2} \)
67 \( 1 - 268065007707894476 T + p^{19} T^{2} \)
71 \( 1 + 113273531338221288 T + p^{19} T^{2} \)
73 \( 1 + 545956267317696358 T + p^{19} T^{2} \)
79 \( 1 + 1807609924990106560 T + p^{19} T^{2} \)
83 \( 1 - 1469958731688321372 T + p^{19} T^{2} \)
89 \( 1 - 2974040568798940170 T + p^{19} T^{2} \)
97 \( 1 + 6925686051327380254 T + p^{19} T^{2} \)
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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

−20.02630734542154439776439081307, −19.03218429709025576673488022458, −17.40484097726205862469378361446, −15.86338483159131384041529839609, −13.22282959306147143546782245974, −10.08651009025494464583031204837, −9.145964965857863111118336554989, −6.89416367334842141651329900066, −2.42178828550413484158341741184, 0, 2.42178828550413484158341741184, 6.89416367334842141651329900066, 9.145964965857863111118336554989, 10.08651009025494464583031204837, 13.22282959306147143546782245974, 15.86338483159131384041529839609, 17.40484097726205862469378361446, 19.03218429709025576673488022458, 20.02630734542154439776439081307

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