Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $-0.240 - 0.970i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.336 + 1.90i)2-s + (−1.20 + 1.24i)3-s + (−1.64 + 0.600i)4-s + (1.09 − 3.00i)5-s + (−2.78 − 1.87i)6-s + (−1.23 + 2.13i)7-s + (0.237 + 0.412i)8-s + (−0.113 − 2.99i)9-s + (6.11 + 1.07i)10-s + (2.11 − 1.21i)11-s + (1.23 − 2.77i)12-s + (1.65 − 1.97i)13-s + (−4.48 − 1.63i)14-s + (2.43 + 4.98i)15-s + (−3.39 + 2.84i)16-s + (−3.75 + 0.662i)17-s + ⋯
L(s)  = 1  + (0.237 + 1.34i)2-s + (−0.693 + 0.720i)3-s + (−0.824 + 0.300i)4-s + (0.489 − 1.34i)5-s + (−1.13 − 0.764i)6-s + (−0.465 + 0.805i)7-s + (0.0841 + 0.145i)8-s + (−0.0379 − 0.999i)9-s + (1.93 + 0.340i)10-s + (0.636 − 0.367i)11-s + (0.355 − 0.801i)12-s + (0.458 − 0.547i)13-s + (−1.19 − 0.435i)14-s + (0.629 + 1.28i)15-s + (−0.848 + 0.712i)16-s + (−0.911 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.240 - 0.970i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ -0.240 - 0.970i)$
$L(1)$  $\approx$  $0.521827 + 0.666699i$
$L(\frac12)$  $\approx$  $0.521827 + 0.666699i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (1.20 - 1.24i)T \)
19 \( 1 + (2.59 + 3.50i)T \)
good2 \( 1 + (-0.336 - 1.90i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.09 + 3.00i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.23 - 2.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.11 + 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.65 + 1.97i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (3.75 - 0.662i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.817 + 2.24i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.705 - 3.99i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.82 + 2.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.08iT - 37T^{2} \)
41 \( 1 + (-4.39 + 3.68i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.86 - 1.40i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (3.04 + 0.536i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.11 - 1.13i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.78 - 10.1i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.932 + 0.339i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (6.74 + 1.18i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.208 - 0.0757i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-12.8 + 10.8i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-2.67 - 3.18i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.45 - 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.53 + 5.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.82 + 0.673i)T + (91.1 - 33.1i)T^{2} \)
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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

−15.73328201605330648608716054749, −14.94215372414671739505659017673, −13.42502103057974021485857239611, −12.44585735830709506307883716520, −10.99605404627654203933875124815, −9.203630303789757661902194064745, −8.636730928620953656082329767772, −6.45025585542269377940027715390, −5.63647688299919809064792959410, −4.54512695195024333992952217598, 2.06958223671315624321030731545, 3.92111462231285298760360254686, 6.35196862728925896401019461513, 7.18059511328272096859792979542, 9.695425325417755796342645046194, 10.76018453194102541336107386591, 11.28427386519896597449273695551, 12.55152190219553934859402573139, 13.52948583981186326489990294613, 14.32654071523159067524741510108

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