↑

Properties

Label 2-770-55.32-c1-0-30
Degree $2$
Conductor $770$
Sign $-0.985 + 0.166i$
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 + (−1.12 + 1.12i)3-s − 1.00i·4-s + (−0.277 − 2.21i)5-s + 1.59i·6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 0.462i·9-s + (−1.76 − 1.37i)10-s + (−2.89 − 1.61i)11-s + (1.12 + 1.12i)12-s + (−2.81 − 2.81i)13-s − 1.00i·14-s + (2.81 + 2.18i)15-s − 1.00·16-s + (−4.15 + 4.15i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.650 + 0.650i)3-s − 0.500i·4-s + (−0.123 − 0.992i)5-s + 0.650i·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.154i·9-s + (−0.558 − 0.434i)10-s + (−0.872 − 0.488i)11-s + (0.325 + 0.325i)12-s + (−0.781 − 0.781i)13-s − 0.267i·14-s + (0.725 + 0.564i)15-s − 0.250·16-s + (−1.00 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.985 + 0.166i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.985 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0546708 - 0.650688i\)
\(L(\frac12)\) \(\approx\) \(0.0546708 - 0.650688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.277 + 2.21i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (2.89 + 1.61i)T \)
good3 \( 1 + (1.12 - 1.12i)T - 3iT^{2} \)
13 \( 1 + (2.81 + 2.81i)T + 13iT^{2} \)
17 \( 1 + (4.15 - 4.15i)T - 17iT^{2} \)
19 \( 1 - 3.57T + 19T^{2} \)
23 \( 1 + (-1.88 + 1.88i)T - 23iT^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 + 9.43T + 31T^{2} \)
37 \( 1 + (4.27 + 4.27i)T + 37iT^{2} \)
41 \( 1 + 4.51iT - 41T^{2} \)
43 \( 1 + (-4.32 - 4.32i)T + 43iT^{2} \)
47 \( 1 + (6.00 + 6.00i)T + 47iT^{2} \)
53 \( 1 + (0.325 - 0.325i)T - 53iT^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 + (2.64 + 2.64i)T + 67iT^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + (4.06 + 4.06i)T + 73iT^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (-5.77 - 5.77i)T + 97iT^{2} \)
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

−10.25088439716487766383308664111, −9.241432818737886630121638160240, −8.287379544612883728847467189924, −7.38822514726272137963606326785, −5.83135576293116242622520694034, −5.21096080228846552917160791103, −4.62833518480242022657989370110, −3.56807661764476769524757272746, −2.05218436307062581830116187303, −0.28226695152412407828512261911, 2.12347383037697979797167810203, 3.23229161497576547089509652136, 4.66010779612191649720537560791, 5.47429971603394481090900401402, 6.48530976866982434814868647014, 7.23717907234118743039407523061, 7.52207928460736469575733015985, 9.005116580781262190103283573522, 9.857276739139054203097646991217, 11.11399199440296157187704664531

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