Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $0.704 + 0.709i$
Motivic weight 16
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 181. i·2-s + (−4.65e3 + 4.62e3i)3-s − 3.27e4·4-s + 1.53e3i·5-s + (8.37e5 + 8.42e5i)6-s + 5.92e6·7-s + 5.93e6i·8-s + (2.74e5 − 4.30e7i)9-s + 2.78e5·10-s − 2.30e8i·11-s + (1.52e8 − 1.51e8i)12-s + 1.44e9·13-s − 1.07e9i·14-s + (−7.12e6 − 7.16e6i)15-s + 1.07e9·16-s + 1.68e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.709 + 0.704i)3-s − 0.500·4-s + 0.00394i·5-s + (0.498 + 0.501i)6-s + 1.02·7-s + 0.353i·8-s + (0.00637 − 0.999i)9-s + 0.00278·10-s − 1.07i·11-s + (0.354 − 0.352i)12-s + 1.77·13-s − 0.726i·14-s + (−0.00277 − 0.00279i)15-s + 0.250·16-s + 0.0241i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $0.704 + 0.709i$
motivic weight  =  \(16\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :8),\ 0.704 + 0.709i)$
$L(\frac{17}{2})$  $\approx$  $1.35830 - 0.565167i$
$L(\frac12)$  $\approx$  $1.35830 - 0.565167i$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + 181. iT \)
3 \( 1 + (4.65e3 - 4.62e3i)T \)
good5 \( 1 - 1.53e3iT - 1.52e11T^{2} \)
7 \( 1 - 5.92e6T + 3.32e13T^{2} \)
11 \( 1 + 2.30e8iT - 4.59e16T^{2} \)
13 \( 1 - 1.44e9T + 6.65e17T^{2} \)
17 \( 1 - 1.68e8iT - 4.86e19T^{2} \)
19 \( 1 - 4.32e9T + 2.88e20T^{2} \)
23 \( 1 - 9.99e10iT - 6.13e21T^{2} \)
29 \( 1 + 7.44e11iT - 2.50e23T^{2} \)
31 \( 1 - 1.99e11T + 7.27e23T^{2} \)
37 \( 1 + 3.98e12T + 1.23e25T^{2} \)
41 \( 1 + 1.34e13iT - 6.37e25T^{2} \)
43 \( 1 - 9.66e12T + 1.36e26T^{2} \)
47 \( 1 + 2.89e13iT - 5.66e26T^{2} \)
53 \( 1 - 8.03e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.63e14iT - 2.15e28T^{2} \)
61 \( 1 - 8.55e13T + 3.67e28T^{2} \)
67 \( 1 - 5.13e14T + 1.64e29T^{2} \)
71 \( 1 - 3.19e14iT - 4.16e29T^{2} \)
73 \( 1 - 2.79e13T + 6.50e29T^{2} \)
79 \( 1 + 4.23e14T + 2.30e30T^{2} \)
83 \( 1 - 6.58e14iT - 5.07e30T^{2} \)
89 \( 1 + 4.19e15iT - 1.54e31T^{2} \)
97 \( 1 + 3.81e15T + 6.14e31T^{2} \)
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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

−18.64412099819644533670884444351, −17.35308933834167011568023550979, −15.70208266118348981396642894426, −13.78306895293030632099182476931, −11.61383018243850042397179835926, −10.73487614945311479726625060650, −8.719216736871403607528844815536, −5.60487274385065624811442874071, −3.76998401104160748206688357072, −1.00393931357795407866733308516, 1.30961991987416287929297389648, 4.89930800873217787276235917993, 6.66888136939713409277169380881, 8.294918459689369405650675662991, 10.95040302868749819873820900422, 12.73226252099503846960440172379, 14.37304997369230373112233152450, 16.18182350457992868871800156748, 17.69935840960165626522504126794, 18.46633285122186018604934690934

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