Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $-0.304 - 0.952i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.45 + 0.934i)3-s + (−1.90 + 3.29i)5-s + (−2.23 − 1.41i)7-s + (1.25 + 2.72i)9-s + (0.309 − 0.178i)11-s + 4.04i·13-s + (−5.84 + 3.02i)15-s + (−0.0519 − 0.0900i)17-s + (2.12 + 1.22i)19-s + (−1.93 − 4.15i)21-s + (−1.15 − 0.665i)23-s + (−4.72 − 8.17i)25-s + (−0.723 + 5.14i)27-s + 4.97i·29-s + (6.83 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.841 + 0.539i)3-s + (−0.849 + 1.47i)5-s + (−0.844 − 0.535i)7-s + (0.417 + 0.908i)9-s + (0.0933 − 0.0538i)11-s + 1.12i·13-s + (−1.50 + 0.780i)15-s + (−0.0126 − 0.0218i)17-s + (0.487 + 0.281i)19-s + (−0.422 − 0.906i)21-s + (−0.240 − 0.138i)23-s + (−0.944 − 1.63i)25-s + (−0.139 + 0.990i)27-s + 0.923i·29-s + (1.22 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.304 - 0.952i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ -0.304 - 0.952i)$
$L(1)$  $\approx$  $0.769182 + 1.05316i$
$L(\frac12)$  $\approx$  $0.769182 + 1.05316i$
$L(\frac{3}{2})$   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 \( 1 + (-1.45 - 0.934i)T \)
7 \( 1 + (2.23 + 1.41i)T \)
good5 \( 1 + (1.90 - 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.309 + 0.178i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.04iT - 13T^{2} \)
17 \( 1 + (0.0519 + 0.0900i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.12 - 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.15 + 0.665i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.97iT - 29T^{2} \)
31 \( 1 + (-6.83 + 3.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.45 + 9.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 + 0.502T + 43T^{2} \)
47 \( 1 + (5.72 - 9.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.08 + 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.77 + 6.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.20 - 4.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 - 2.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.78iT - 71T^{2} \)
73 \( 1 + (0.203 - 0.117i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.61 + 2.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 + (-3.41 + 5.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−11.55279525522866921362255347803, −10.82972185319706724503133838808, −9.982538972809937611340927580644, −9.214167886608814393697660751836, −7.88700365179089065986067197980, −7.19508377831891081668701020108, −6.30327243512666475269033143958, −4.31090868008685798545777892672, −3.57692607484081581047623259910, −2.59526906863854710148238366343, 0.871030348033655795552330098128, 2.81393952623480763804756503467, 3.96995128990737407099586955054, 5.25127195187350083973448930200, 6.52808451544070357470227451900, 7.85477903362729193107898077611, 8.336009866331646022460798534154, 9.232027856707322671390029697869, 10.00833320195499414705542666606, 11.79645280889782524004423416912

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