Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $0.999 - 0.000124i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 − 1.22i)3-s + (−1.40 + 2.43i)5-s + (2.08 + 1.62i)7-s + (−0.0238 − 2.99i)9-s + (4.74 − 2.74i)11-s + 1.35i·13-s + (1.27 + 4.69i)15-s + (2.88 + 5.00i)17-s + (−1.71 − 0.992i)19-s + (4.54 − 0.579i)21-s + (−2.09 − 1.21i)23-s + (−1.44 − 2.49i)25-s + (−3.71 − 3.63i)27-s − 7.05i·29-s + (3.07 − 1.77i)31-s + ⋯
L(s)  = 1  + (0.704 − 0.709i)3-s + (−0.627 + 1.08i)5-s + (0.788 + 0.615i)7-s + (−0.00795 − 0.999i)9-s + (1.43 − 0.826i)11-s + 0.376i·13-s + (0.329 + 1.21i)15-s + (0.700 + 1.21i)17-s + (−0.394 − 0.227i)19-s + (0.991 − 0.126i)21-s + (−0.437 − 0.252i)23-s + (−0.288 − 0.499i)25-s + (−0.715 − 0.698i)27-s − 1.31i·29-s + (0.552 − 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.000124i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ 0.999 - 0.000124i)$
$L(1)$  $\approx$  $1.68490 + 0.000104851i$
$L(\frac12)$  $\approx$  $1.68490 + 0.000104851i$
$L(\frac{3}{2})$   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 \( 1 + (-1.21 + 1.22i)T \)
7 \( 1 + (-2.08 - 1.62i)T \)
good5 \( 1 + (1.40 - 2.43i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.74 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.71 + 0.992i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.09 + 1.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.05iT - 29T^{2} \)
31 \( 1 + (-3.07 + 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.14 - 3.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.81T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (-0.201 + 0.348i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.28 - 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.28 + 2.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.75 + 2.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.45 + 5.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + (0.295 - 0.170i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.19 - 2.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (0.576 - 0.998i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−11.74843076488367753402519792953, −10.85198988349923257001671848582, −9.542327867340186099285504712918, −8.423524644253429013761306770640, −7.977986157760903711391396429246, −6.69892168033533389524293677377, −6.10924955841441549340211703221, −4.10536085983246885683666185475, −3.14953301750493780611651477647, −1.71651202358446154606198578513, 1.48796186236915409679844702052, 3.52591270455430819486177588856, 4.48331063650910277451485600158, 5.10975409359444917876040952996, 7.06255956338600053682262677629, 7.973160725232823749244837395493, 8.759016344798610593559738223117, 9.566244413399635868944938173884, 10.49931033033028602131759383053, 11.67962292133190185009537363376

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