Properties

Label 2-770-35.27-c1-0-38
Degree $2$
Conductor $770$
Sign $-0.966 + 0.258i$
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  + (−0.707 − 0.707i)2-s + (0.839 + 0.839i)3-s + 1.00i·4-s + (−0.690 − 2.12i)5-s − 1.18i·6-s + (−0.472 − 2.60i)7-s + (0.707 − 0.707i)8-s − 1.59i·9-s + (−1.01 + 1.99i)10-s − 11-s + (−0.839 + 0.839i)12-s + (−0.602 − 0.602i)13-s + (−1.50 + 2.17i)14-s + (1.20 − 2.36i)15-s − 1.00·16-s + (−2.58 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.484 + 0.484i)3-s + 0.500i·4-s + (−0.308 − 0.951i)5-s − 0.484i·6-s + (−0.178 − 0.983i)7-s + (0.250 − 0.250i)8-s − 0.530i·9-s + (−0.321 + 0.629i)10-s − 0.301·11-s + (−0.242 + 0.242i)12-s + (−0.167 − 0.167i)13-s + (−0.402 + 0.581i)14-s + (0.311 − 0.610i)15-s − 0.250·16-s + (−0.626 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.966 + 0.258i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.966 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0874961 - 0.665557i\)
\(L(\frac12)\) \(\approx\) \(0.0874961 - 0.665557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.690 + 2.12i)T \)
7 \( 1 + (0.472 + 2.60i)T \)
11 \( 1 + T \)
good3 \( 1 + (-0.839 - 0.839i)T + 3iT^{2} \)
13 \( 1 + (0.602 + 0.602i)T + 13iT^{2} \)
17 \( 1 + (2.58 - 2.58i)T - 17iT^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 + (4.92 - 4.92i)T - 23iT^{2} \)
29 \( 1 - 6.57iT - 29T^{2} \)
31 \( 1 + 5.10iT - 31T^{2} \)
37 \( 1 + (5.69 + 5.69i)T + 37iT^{2} \)
41 \( 1 + 8.40iT - 41T^{2} \)
43 \( 1 + (6.41 - 6.41i)T - 43iT^{2} \)
47 \( 1 + (-3.53 + 3.53i)T - 47iT^{2} \)
53 \( 1 + (-6.49 + 6.49i)T - 53iT^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 + (6.87 + 6.87i)T + 67iT^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + (3.43 + 3.43i)T + 73iT^{2} \)
79 \( 1 - 1.87iT - 79T^{2} \)
83 \( 1 + (3.37 + 3.37i)T + 83iT^{2} \)
89 \( 1 + 5.67T + 89T^{2} \)
97 \( 1 + (-2.87 + 2.87i)T - 97iT^{2} \)
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

−9.894234190052637773772068565364, −9.121911066711273693868932429826, −8.427509723659231274360458181386, −7.65752996208996958202734735173, −6.60446870033565806128290915259, −5.19357485205892454663828904697, −3.97027922723059836554886099529, −3.64246578370944630850394128479, −1.90966691560183084339889919051, −0.35456766151931119635228246056, 2.15285437367240067330702637153, 2.78133665363837657460690477177, 4.43653065136752094813708750091, 5.66103977913937673823276240892, 6.63123284961080043526117260643, 7.22204644438466081988783496637, 8.324250852275126933277048413363, 8.575978101778573274201987867547, 9.883641747453252919563115368746, 10.48471294988103250741123918855

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