Properties

Label 2-770-11.5-c1-0-1
Degree $2$
Conductor $770$
Sign $-0.520 - 0.854i$
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  + (0.309 + 0.951i)2-s + (−0.988 − 0.718i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.377 − 1.16i)6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.465 − 1.43i)9-s − 0.999·10-s + (0.912 + 3.18i)11-s + 1.22·12-s + (1.07 + 3.30i)13-s + (0.809 + 0.587i)14-s + (0.988 − 0.718i)15-s + (0.309 − 0.951i)16-s + (−0.0426 + 0.131i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.570 − 0.414i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.154 − 0.474i)6-s + (0.305 − 0.222i)7-s + (−0.286 − 0.207i)8-s + (−0.155 − 0.477i)9-s − 0.316·10-s + (0.275 + 0.961i)11-s + 0.352·12-s + (0.298 + 0.917i)13-s + (0.216 + 0.157i)14-s + (0.255 − 0.185i)15-s + (0.0772 − 0.237i)16-s + (−0.0103 + 0.0318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.520 - 0.854i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.520 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518004 + 0.922015i\)
\(L(\frac12)\) \(\approx\) \(0.518004 + 0.922015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.912 - 3.18i)T \)
good3 \( 1 + (0.988 + 0.718i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-1.07 - 3.30i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.0426 - 0.131i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.416 + 0.302i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 + (7.47 - 5.43i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.17 - 9.76i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.34 - 0.980i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-9.85 - 7.16i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + (-1.46 - 1.06i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.429 - 1.32i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.52 - 5.46i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.17 + 12.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.84T + 67T^{2} \)
71 \( 1 + (1.97 - 6.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.95 + 6.50i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.08 - 9.49i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.681 + 2.09i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 + (1.36 + 4.19i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.77034781950531178946343521422, −9.582687003331928414302947373894, −8.875108978157609608491280363325, −7.74658595918182845860486875964, −6.81837199217864207185835132668, −6.57728963165216088923976321820, −5.34972820665433701902802796698, −4.41767912060930473180648286443, −3.33022078604293612375010905937, −1.53653739108378959345099883193, 0.56142577142602890964145999971, 2.24704491706243020660887805469, 3.58796953897119792338895610933, 4.52732329127100457168945005907, 5.56486691183356078082917540166, 5.96195702718650701166723787858, 7.70126361282279919932532806676, 8.411660750783145768672465577476, 9.336642783432106625865104423947, 10.23332601561377900624150968246

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