Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $-0.761 + 0.648i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.291 − 1.70i)3-s + (−0.0726 − 0.125i)5-s + (−1.05 − 2.42i)7-s + (−2.83 + 0.993i)9-s + (−2.13 − 1.23i)11-s − 2.04i·13-s + (−0.193 + 0.160i)15-s + (−0.878 + 1.52i)17-s + (3.68 − 2.12i)19-s + (−3.83 + 2.50i)21-s + (−7.46 + 4.30i)23-s + (2.48 − 4.31i)25-s + (2.52 + 4.54i)27-s − 7.08i·29-s + (−3.11 − 1.80i)31-s + ⋯
L(s)  = 1  + (−0.168 − 0.985i)3-s + (−0.0324 − 0.0562i)5-s + (−0.398 − 0.917i)7-s + (−0.943 + 0.331i)9-s + (−0.644 − 0.372i)11-s − 0.566i·13-s + (−0.0500 + 0.0414i)15-s + (−0.213 + 0.369i)17-s + (0.844 − 0.487i)19-s + (−0.837 + 0.547i)21-s + (−1.55 + 0.898i)23-s + (0.497 − 0.862i)25-s + (0.485 + 0.874i)27-s − 1.31i·29-s + (−0.560 − 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.761 + 0.648i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ -0.761 + 0.648i)$
$L(1)$  $\approx$  $0.307460 - 0.835089i$
$L(\frac12)$  $\approx$  $0.307460 - 0.835089i$
$L(\frac{3}{2})$   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 \( 1 + (0.291 + 1.70i)T \)
7 \( 1 + (1.05 + 2.42i)T \)
good5 \( 1 + (0.0726 + 0.125i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.13 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + (0.878 - 1.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.46 - 4.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.08iT - 29T^{2} \)
31 \( 1 + (3.11 + 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.93 + 5.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.33T + 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 + (-4.65 - 8.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.49 - 2.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.60 + 9.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.66 + 2.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.57 - 4.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.79iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.86 + 4.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + (4.34 + 7.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.65iT - 97T^{2} \)
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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

−11.19801952038849350557748522925, −10.44767779959031999919183804526, −9.337560665827352523029567476376, −7.946851016238225064591270584515, −7.56920556763165811660658265506, −6.34293036879244322366249347218, −5.50003987318121323270921185457, −3.90561377274319889789095507235, −2.47679164326415729296126314633, −0.62198921798805831131566549335, 2.50359363019623752521709464954, 3.75190071243793147629420458520, 5.04341257435757499167842880092, 5.80593888074789927672800900378, 7.06374366302601765782958849424, 8.446733878366181443230288385003, 9.227519341173956965770124105683, 10.03075965079440109384462994246, 10.88142895044952584466169901552, 11.92016963096340527460188416320

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