Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.0300 + 0.999i$
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 + (1.43 − 0.966i)3-s + (−1.64 + 0.600i)4-s + (−1.09 + 3.00i)5-s + (−2.32 − 2.41i)6-s + (−1.23 + 2.13i)7-s + (−0.237 − 0.412i)8-s + (1.13 − 2.77i)9-s + (6.11 + 1.07i)10-s + (−2.11 + 1.21i)11-s + (−1.78 + 2.45i)12-s + (1.65 − 1.97i)13-s + (4.48 + 1.63i)14-s + (1.33 + 5.38i)15-s + (−3.39 + 2.84i)16-s + (3.75 − 0.662i)17-s + ⋯
L(s)  = 1  + (−0.237 − 1.34i)2-s + (0.829 − 0.557i)3-s + (−0.824 + 0.300i)4-s + (−0.489 + 1.34i)5-s + (−0.950 − 0.987i)6-s + (−0.465 + 0.805i)7-s + (−0.0841 − 0.145i)8-s + (0.377 − 0.926i)9-s + (1.93 + 0.340i)10-s + (−0.636 + 0.367i)11-s + (−0.516 + 0.708i)12-s + (0.458 − 0.547i)13-s + (1.19 + 0.435i)14-s + (0.344 + 1.39i)15-s + (−0.848 + 0.712i)16-s + (0.911 − 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.0300 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.0300 + 0.999i)$
$L(1)$  $\approx$  $0.628502 - 0.609883i$
$L(\frac12)$  $\approx$  $0.628502 - 0.609883i$
$L(\frac{3}{2})$   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 \( 1 + (-1.43 + 0.966i)T \)
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

−15.05425671955109804187837604095, −13.51553875560096729367752359799, −12.55881422783037588777666512152, −11.55401989751276207136648704957, −10.41909379196453081170428931386, −9.370464394386344758473488962986, −7.892575599594483906500224674935, −6.51189619201459778390899520089, −3.38830114165394806824490771880, −2.54586146430352091830856281159, 3.99407877930334346704347257383, 5.43594230161091345774193366818, 7.32504642368223883126646029561, 8.342886032887341889969256522064, 9.056572118784488988334963947531, 10.55132057614183621244177802120, 12.50746277755758118646665997586, 13.67562729205643419634523263233, 14.62687094278187924858535298065, 15.86205764318458976216103838745

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