Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.448 - 0.893i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 + 3.94i)2-s + (1.5 − 2.59i)3-s + (−6.38 + 11.0i)4-s + (−8.93 − 15.4i)5-s + 13.6·6-s + (2.26 + 18.3i)7-s − 21.6·8-s + (−4.5 − 7.79i)9-s + (40.7 − 70.5i)10-s + (5.69 − 9.86i)11-s + (19.1 + 33.1i)12-s − 13.0·13-s + (−67.3 + 50.7i)14-s − 53.6·15-s + (1.62 + 2.81i)16-s + (−26.6 + 46.1i)17-s + ⋯
L(s)  = 1  + (0.805 + 1.39i)2-s + (0.288 − 0.499i)3-s + (−0.797 + 1.38i)4-s + (−0.799 − 1.38i)5-s + 0.930·6-s + (0.122 + 0.992i)7-s − 0.958·8-s + (−0.166 − 0.288i)9-s + (1.28 − 2.23i)10-s + (0.156 − 0.270i)11-s + (0.460 + 0.797i)12-s − 0.279·13-s + (−1.28 + 0.969i)14-s − 0.922·15-s + (0.0254 + 0.0440i)16-s + (−0.379 + 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.448 - 0.893i$
motivic weight  =  \(3\)
character  :  $\chi_{21} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :3/2),\ 0.448 - 0.893i)$
$L(2)$  $\approx$  $1.27535 + 0.787359i$
$L(\frac12)$  $\approx$  $1.27535 + 0.787359i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (-2.26 - 18.3i)T \)
good2 \( 1 + (-2.27 - 3.94i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (8.93 + 15.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-5.69 + 9.86i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 + (26.6 - 46.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-21.2 - 36.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (76.0 + 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 186.T + 2.43e4T^{2} \)
31 \( 1 + (-78.9 + 136. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (1.87 + 3.24i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + (10.5 + 18.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (182. - 316. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (325. + 564. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (72.7 - 125. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 368.T + 3.57e5T^{2} \)
73 \( 1 + (304. - 527. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (455. + 788. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 327.T + 5.71e5T^{2} \)
89 \( 1 + (-18.8 - 32.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 722.T + 9.12e5T^{2} \)
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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

−17.43774137561550892199505646202, −16.27073886035681434729160253309, −15.46422482729174497083026037399, −14.25503095150408340233960736069, −12.82057622668740306790035699203, −12.10326828635448701727356668854, −8.722991757986355289959678706162, −7.952594403274095311263756886792, −6.04758516490262056537973288899, −4.49089376306427944573774469808, 3.05831546450298624563908847700, 4.35379303846738909285594228489, 7.32358574739059563002668662537, 9.974220816746377401584436604345, 10.90973191375250389061564735008, 11.84060259256191149626344280730, 13.69630978552099495546249402480, 14.44833881085257986104330674284, 15.76998630492813507406694678570, 17.85938090637667981523434693904

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