Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.921 - 0.388i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.08 + 0.759i)2-s + (−1.69 − 0.352i)3-s + (2.24 + 1.88i)4-s + (−1.78 − 2.12i)5-s + (−3.26 − 2.02i)6-s + (−1.70 + 2.96i)7-s + (1.02 + 1.77i)8-s + (2.75 + 1.19i)9-s + (−2.10 − 5.77i)10-s + (2.03 − 1.17i)11-s + (−3.13 − 3.97i)12-s + (−0.115 − 0.0203i)13-s + (−5.81 + 4.87i)14-s + (2.27 + 4.22i)15-s + (−0.224 − 1.27i)16-s + (0.519 − 1.42i)17-s + ⋯
L(s)  = 1  + (1.47 + 0.536i)2-s + (−0.979 − 0.203i)3-s + (1.12 + 0.940i)4-s + (−0.796 − 0.948i)5-s + (−1.33 − 0.825i)6-s + (−0.645 + 1.11i)7-s + (0.363 + 0.629i)8-s + (0.917 + 0.398i)9-s + (−0.664 − 1.82i)10-s + (0.614 − 0.354i)11-s + (−0.905 − 1.14i)12-s + (−0.0320 − 0.00565i)13-s + (−1.55 + 1.30i)14-s + (0.586 + 1.09i)15-s + (−0.0560 − 0.318i)16-s + (0.125 − 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.921 - 0.388i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.921 - 0.388i)$
$L(1)$  $\approx$  $1.17354 + 0.237451i$
$L(\frac12)$  $\approx$  $1.17354 + 0.237451i$
$L(\frac{3}{2})$   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 \( 1 + (1.69 + 0.352i)T \)
19 \( 1 + (-3.41 - 2.71i)T \)
good2 \( 1 + (-2.08 - 0.759i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (1.78 + 2.12i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.70 - 2.96i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.03 + 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.115 + 0.0203i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.519 + 1.42i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.79 - 5.71i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.69 - 0.616i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.23 + 0.711i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.31iT - 37T^{2} \)
41 \( 1 + (0.635 + 3.60i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.60 + 3.02i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.61 - 7.17i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.93 + 2.46i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (10.5 + 3.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.58 - 5.52i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.0605 + 0.166i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.60 + 4.70i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.838 - 4.75i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (6.89 - 1.21i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.836 + 0.482i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.02 - 11.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.73 + 4.75i)T + (-74.3 - 62.3i)T^{2} \)
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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

−15.76836835510424768909031125918, −14.09810162838140733651240495952, −12.83757219089117642718502421960, −12.15779147844061998267471414961, −11.61740818054922512919344060039, −9.383499882706547280546874784133, −7.58670926006945507886528192891, −6.08053631408562331209743595811, −5.29459224320675513514345438569, −3.85511470514252493513529143072, 3.51545549577397121983584028476, 4.47002249661510748163663902597, 6.26333670625796508285792905655, 7.18664471530132462726369639029, 10.08395826306920917869627776775, 10.98965456258373081085972763671, 11.81488372679682478988301557221, 12.78285933279570675689484210962, 13.97058748702675437390429061878, 14.97744673581358583674387429458

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