Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $-0.340 - 0.940i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 + 0.247i)3-s + (−1.28 − 2.23i)5-s + (0.203 + 2.63i)7-s + (2.87 − 0.849i)9-s + (−1.43 − 0.826i)11-s + 5.71i·13-s + (2.76 + 3.50i)15-s + (−3.79 + 6.56i)17-s + (−2.58 + 1.49i)19-s + (−1.00 − 4.47i)21-s + (−0.249 + 0.143i)23-s + (−0.825 + 1.43i)25-s + (−4.72 + 2.16i)27-s + 2.05i·29-s + (5.21 + 3.00i)31-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)3-s + (−0.576 − 0.998i)5-s + (0.0768 + 0.997i)7-s + (0.959 − 0.283i)9-s + (−0.431 − 0.249i)11-s + 1.58i·13-s + (0.713 + 0.906i)15-s + (−0.919 + 1.59i)17-s + (−0.594 + 0.343i)19-s + (−0.218 − 0.975i)21-s + (−0.0519 + 0.0300i)23-s + (−0.165 + 0.286i)25-s + (−0.908 + 0.417i)27-s + 0.382i·29-s + (0.936 + 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.340 - 0.940i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ -0.340 - 0.940i)$
$L(1)$  $\approx$  $0.298031 + 0.424775i$
$L(\frac12)$  $\approx$  $0.298031 + 0.424775i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.71 - 0.247i)T \)
7 \( 1 + (-0.203 - 2.63i)T \)
good5 \( 1 + (1.28 + 2.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.43 + 0.826i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.71iT - 13T^{2} \)
17 \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 1.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.249 - 0.143i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (-5.21 - 3.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.877 + 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 + (0.186 + 0.323i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.73 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.89 + 8.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.889 + 0.513i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.18 + 2.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (3.30 + 1.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.56 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + (7.25 + 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.43iT - 97T^{2} \)
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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

−11.86755765721120124920275732567, −11.14214244405160153665502839864, −10.08697648702122049970994543601, −8.828361022634233497811527458369, −8.393535373930653808480382131015, −6.79767715010268127192566601734, −5.92974577836550975265984875506, −4.82541547100784576816787901012, −4.06445137388215299026409577301, −1.78492664648249292658765246270, 0.40899089889367479778663284974, 2.80991057042183285128581666437, 4.24727655174244728373855025945, 5.28524573265886313781737532483, 6.66930395429974060767042098821, 7.22957120670664213153124449574, 8.084414866485772857316186549054, 9.878827095681360185077198554248, 10.55388865373928860501196450289, 11.15991444158966571747683969393

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