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  :  $\chi_{57} (14, \cdot )$
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

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

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