↑

Properties

Label 2-1014-39.8-c1-0-38
Degree $2$
Conductor $1014$
Sign $-0.756 + 0.654i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.58 − 0.688i)3-s − 1.00i·4-s + (−0.822 + 0.822i)5-s + (−0.637 + 1.61i)6-s + (−1.88 + 1.88i)7-s + (0.707 + 0.707i)8-s + (2.05 − 2.18i)9-s − 1.16i·10-s + (−3.44 − 3.44i)11-s + (−0.688 − 1.58i)12-s − 2.66i·14-s + (−0.741 + 1.87i)15-s − 1.00·16-s − 2.82·17-s + (0.0946 + 2.99i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.917 − 0.397i)3-s − 0.500i·4-s + (−0.367 + 0.367i)5-s + (−0.260 + 0.657i)6-s + (−0.711 + 0.711i)7-s + (0.250 + 0.250i)8-s + (0.684 − 0.729i)9-s − 0.367i·10-s + (−1.03 − 1.03i)11-s + (−0.198 − 0.458i)12-s − 0.711i·14-s + (−0.191 + 0.483i)15-s − 0.250·16-s − 0.684·17-s + (0.0223 + 0.706i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.756 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2721916167\)
\(L(\frac12)\) \(\approx\) \(0.2721916167\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.58 + 0.688i)T \)
13 \( 1 \)
good5 \( 1 + (0.822 - 0.822i)T - 5iT^{2} \)
7 \( 1 + (1.88 - 1.88i)T - 7iT^{2} \)
11 \( 1 + (3.44 + 3.44i)T + 11iT^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (3.54 + 3.54i)T + 19iT^{2} \)
23 \( 1 + 9.31T + 23T^{2} \)
29 \( 1 - 3.13iT - 29T^{2} \)
31 \( 1 + (0.709 + 0.709i)T + 31iT^{2} \)
37 \( 1 + (-2.63 + 2.63i)T - 37iT^{2} \)
41 \( 1 + (-3.75 + 3.75i)T - 41iT^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 + (-8.45 - 8.45i)T + 47iT^{2} \)
53 \( 1 + 1.57iT - 53T^{2} \)
59 \( 1 + (3.86 + 3.86i)T + 59iT^{2} \)
61 \( 1 + 9.76T + 61T^{2} \)
67 \( 1 + (7.61 + 7.61i)T + 67iT^{2} \)
71 \( 1 + (0.592 - 0.592i)T - 71iT^{2} \)
73 \( 1 + (6.42 - 6.42i)T - 73iT^{2} \)
79 \( 1 + 5.98T + 79T^{2} \)
83 \( 1 + (1.75 - 1.75i)T - 83iT^{2} \)
89 \( 1 + (4.76 + 4.76i)T + 89iT^{2} \)
97 \( 1 + (-7.52 - 7.52i)T + 97iT^{2} \)
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

−9.215962509883402158429608194922, −8.850472386256571398767952906686, −7.967926570119731782134662227739, −7.33004598824561512257053695111, −6.34989312105441367978025984390, −5.68543200938733000443918553994, −4.15265467537723235296230737821, −3.01063733247415139521536703917, −2.18320951239707721390412665327, −0.11857642095868511636596159081, 1.91090815884163327262875869522, 2.85670281425326118165201995325, 4.18990194822037336501041933632, 4.39220666099151092579599358478, 6.14891175844707281973110721933, 7.37495893832256409805384263123, 7.930288047472742206088824150489, 8.610883881804532978639912783819, 9.671178977012982187811120045176, 10.14832168196153334341129202700

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