Properties

Degree $2$
Conductor $1008$
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

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.213 + 0.976i$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.948538291\)
\(L(\frac12)\) \(\approx\) \(1.948538291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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