Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.213 - 0.976i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 + 1.14i)3-s + (−0.164 + 0.284i)5-s + (−0.5 − 0.866i)7-s + (0.400 + 2.97i)9-s + (−0.664 − 1.15i)11-s + (−1.53 + 2.66i)13-s + (−0.539 + 0.183i)15-s + 7.35·17-s + 2.93·19-s + (0.335 − 1.69i)21-s + (−3.34 + 5.79i)23-s + (2.44 + 4.23i)25-s + (−2.86 + 4.33i)27-s + (3.88 + 6.72i)29-s + (−1.63 + 2.83i)31-s + ⋯
L(s)  = 1  + (0.752 + 0.658i)3-s + (−0.0735 + 0.127i)5-s + (−0.188 − 0.327i)7-s + (0.133 + 0.991i)9-s + (−0.200 − 0.347i)11-s + (−0.426 + 0.739i)13-s + (−0.139 + 0.0474i)15-s + 1.78·17-s + 0.673·19-s + (0.0732 − 0.370i)21-s + (−0.697 + 1.20i)23-s + (0.489 + 0.847i)25-s + (−0.552 + 0.833i)27-s + (0.720 + 1.24i)29-s + (−0.293 + 0.508i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.213 - 0.976i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.213 - 0.976i)\)
\(L(1)\)  \(\approx\)  \(1.948538291\)
\(L(\frac12)\)  \(\approx\)  \(1.948538291\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.30 - 1.14i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.164 - 0.284i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.664 + 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.53 - 2.66i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.35T + 17T^{2} \)
19 \( 1 - 2.93T + 19T^{2} \)
23 \( 1 + (3.34 - 5.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.88 - 6.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.329T + 37T^{2} \)
41 \( 1 + (0.135 - 0.234i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.48 + 9.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.571 - 0.989i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (-0.372 + 0.644i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.42 + 7.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.28 + 7.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.60T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + (0.628 + 1.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0316 + 0.0548i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.03451416320941426275088265751, −9.408911307024673306823225355852, −8.578379941677927568701620964899, −7.61946230131509330652176286098, −7.08795708857482909548177646409, −5.61334109586555264927880977765, −4.91301900421821281093291319273, −3.59688393453420917764806683674, −3.16723053330200251047595758924, −1.59768821872599677791134235527, 0.873282736670917357803788238461, 2.40984997072479603826990605551, 3.16200537274495010437109373348, 4.39560705230468956801808294129, 5.61263629115377492106811649710, 6.42608376166459075922207433109, 7.55695302995748684866261387600, 7.995165182638250427554117296378, 8.799457302156853025994683663389, 9.986183769993446895467171635388

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