↑

Properties

Label 2-770-7.2-c1-0-0
Degree $2$
Conductor $770$
Sign $-0.563 + 0.826i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (−0.5 + 0.866i)2-s + (1.53 + 2.66i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 3.07·6-s + (−2.08 − 1.62i)7-s + 0.999·8-s + (−3.22 + 5.59i)9-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (1.53 − 2.66i)12-s − 4.72·13-s + (2.45 − 0.993i)14-s − 3.07·15-s + (−0.5 + 0.866i)16-s + (−0.549 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.887 + 1.53i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 1.25·6-s + (−0.788 − 0.614i)7-s + 0.353·8-s + (−1.07 + 1.86i)9-s + (−0.158 − 0.273i)10-s + (−0.150 − 0.261i)11-s + (0.443 − 0.768i)12-s − 1.31·13-s + (0.655 − 0.265i)14-s − 0.794·15-s + (−0.125 + 0.216i)16-s + (−0.133 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.563 + 0.826i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317593 - 0.601148i\)
\(L(\frac12)\) \(\approx\) \(0.317593 - 0.601148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.53 - 2.66i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 4.72T + 13T^{2} \)
17 \( 1 + (0.549 + 0.951i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.19 - 2.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0856 - 0.148i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 + (-0.190 - 0.330i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.35 - 9.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 - 3.82T + 43T^{2} \)
47 \( 1 + (2.38 - 4.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.36 - 7.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.81 + 4.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.07 + 8.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.72T + 71T^{2} \)
73 \( 1 + (-7.50 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.51T + 83T^{2} \)
89 \( 1 + (-4.34 + 7.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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

−10.43271727133880264407054836754, −9.858455122347172625382495174532, −9.372821201603443539735411710132, −8.381321096252218015144844768913, −7.60877404431605476190193099021, −6.69049201709167798948428531033, −5.40182190219259142295402678593, −4.46454240707140301213818005431, −3.59219601509572126803160446504, −2.62200247349415929642191656418, 0.32015030768246258855828400863, 1.99720126832606614787343787633, 2.61770210167950801205196542829, 3.76358851857842878296958497737, 5.33231127538193881633198889921, 6.61510074221210801007101506862, 7.34909208594914856210967636738, 8.056125193516281362606824926379, 9.034063557360800539321955551558, 9.352856293497799699546372308987

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