Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $-0.585 + 0.810i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.12i·2-s + (−5.04 − 1.22i)3-s − 9·4-s + 10.0·5-s + (−5.04 + 20.8i)6-s + (7 − 17.1i)7-s + 4.12i·8-s + (23.9 + 12.3i)9-s − 41.6i·10-s + 32.9i·11-s + (45.4 + 11.0i)12-s + 56.3i·13-s + (−70.6 − 28.8i)14-s + (−50.9 − 12.3i)15-s − 55·16-s + 60.5·17-s + ⋯
L(s)  = 1  − 1.45i·2-s + (−0.971 − 0.235i)3-s − 1.12·4-s + 0.903·5-s + (−0.343 + 1.41i)6-s + (0.377 − 0.925i)7-s + 0.182i·8-s + (0.888 + 0.458i)9-s − 1.31i·10-s + 0.904i·11-s + (1.09 + 0.265i)12-s + 1.20i·13-s + (−1.34 − 0.550i)14-s + (−0.877 − 0.212i)15-s − 0.859·16-s + 0.864·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $-0.585 + 0.810i$
motivic weight  =  \(3\)
character  :  $\chi_{21} (20, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :3/2),\ -0.585 + 0.810i)$
$L(2)$  $\approx$  $0.434740 - 0.850301i$
$L(\frac12)$  $\approx$  $0.434740 - 0.850301i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (5.04 + 1.22i)T \)
7 \( 1 + (-7 + 17.1i)T \)
good2 \( 1 + 4.12iT - 8T^{2} \)
5 \( 1 - 10.0T + 125T^{2} \)
11 \( 1 - 32.9iT - 1.33e3T^{2} \)
13 \( 1 - 56.3iT - 2.19e3T^{2} \)
17 \( 1 - 60.5T + 4.91e3T^{2} \)
19 \( 1 + 36.7iT - 6.85e3T^{2} \)
23 \( 1 + 90.7iT - 1.21e4T^{2} \)
29 \( 1 - 57.7iT - 2.43e4T^{2} \)
31 \( 1 - 254. iT - 2.97e4T^{2} \)
37 \( 1 - 230T + 5.06e4T^{2} \)
41 \( 1 + 141.T + 6.89e4T^{2} \)
43 \( 1 - 44T + 7.95e4T^{2} \)
47 \( 1 + 343.T + 1.03e5T^{2} \)
53 \( 1 - 206. iT - 1.48e5T^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 + 71.0iT - 2.26e5T^{2} \)
67 \( 1 + 64T + 3.00e5T^{2} \)
71 \( 1 + 461. iT - 3.57e5T^{2} \)
73 \( 1 + 88.1iT - 3.89e5T^{2} \)
79 \( 1 + 442T + 4.93e5T^{2} \)
83 \( 1 + 494.T + 5.71e5T^{2} \)
89 \( 1 + 484.T + 7.04e5T^{2} \)
97 \( 1 + 1.09e3iT - 9.12e5T^{2} \)
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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

−17.58815757562798554271153376215, −16.47754962465419592725348151523, −14.09922365050688358506403851190, −12.92106877133152773390542059963, −11.79678700743563857217526952473, −10.60292365581292974983140671336, −9.711212012730355163794789098556, −6.85989490220117982685325390528, −4.59500727037303649384588985717, −1.58894507498030896984846429630, 5.49719705219397201986108773057, 5.96506418283386290183573186618, 7.995433783463858024492288216964, 9.727412047358681193722752408225, 11.48775398808851655567942659096, 13.21969591693071097044109254109, 14.72928974460140868672543666594, 15.74438006503690908624530442426, 16.84385922248274904754310560637, 17.69349989844536787020082450207

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