Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $0.834 - 0.551i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 1.60i)3-s + (1.28 + 2.23i)5-s + (0.203 + 2.63i)7-s + (−2.17 + 2.06i)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 + (4.11 − 2.02i)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  + (−0.371 − 0.928i)3-s + (0.576 + 0.998i)5-s + (0.0768 + 0.997i)7-s + (−0.724 + 0.689i)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 + (0.897 − 0.441i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.834 - 0.551i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ 0.834 - 0.551i)$
$L(1)$  $\approx$  $1.20566 + 0.362307i$
$L(\frac12)$  $\approx$  $1.20566 + 0.362307i$
$L(\frac{3}{2})$   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 \( 1 + (0.642 + 1.60i)T \)
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

−11.86962544232712801783867868299, −10.93109034766312146794818989059, −9.740004347832950164245090735814, −8.898115150053610575015434221755, −7.61365351196521109246471704978, −6.65422193562869872585556181848, −6.11682495291297444976858960661, −4.86242368697793892696606577149, −2.86845563748581082415577490268, −1.87435532066942530968591045522, 1.02123321135000902402846673603, 3.39679287658090198475772577259, 4.45292486738036702010474447560, 5.46128568045865471165382811365, 6.28197980877308278748843154787, 7.947451140087494299677852387073, 8.718999703460147405953944950956, 9.871599355052059887123990961276, 10.38000141971606750985034350488, 11.22811227788553747382750907880

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