Properties

Label 2-770-385.3-c1-0-15
Degree $2$
Conductor $770$
Sign $0.952 - 0.304i$
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  + (−0.933 + 0.358i)2-s + (0.907 − 0.589i)3-s + (0.743 − 0.669i)4-s + (−1.48 − 1.66i)5-s + (−0.635 + 0.875i)6-s + (−0.0897 + 2.64i)7-s + (−0.453 + 0.891i)8-s + (−0.744 + 1.67i)9-s + (1.98 + 1.02i)10-s + (3.16 + 0.980i)11-s + (0.280 − 1.04i)12-s + (0.567 − 3.58i)13-s + (−0.863 − 2.50i)14-s + (−2.33 − 0.638i)15-s + (0.104 − 0.994i)16-s + (3.15 + 1.21i)17-s + ⋯
L(s)  = 1  + (−0.660 + 0.253i)2-s + (0.523 − 0.340i)3-s + (0.371 − 0.334i)4-s + (−0.665 − 0.746i)5-s + (−0.259 + 0.357i)6-s + (−0.0339 + 0.999i)7-s + (−0.160 + 0.315i)8-s + (−0.248 + 0.557i)9-s + (0.628 + 0.324i)10-s + (0.955 + 0.295i)11-s + (0.0808 − 0.301i)12-s + (0.157 − 0.993i)13-s + (−0.230 − 0.668i)14-s + (−0.602 − 0.164i)15-s + (0.0261 − 0.248i)16-s + (0.765 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22177 + 0.190636i\)
\(L(\frac12)\) \(\approx\) \(1.22177 + 0.190636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.933 - 0.358i)T \)
5 \( 1 + (1.48 + 1.66i)T \)
7 \( 1 + (0.0897 - 2.64i)T \)
11 \( 1 + (-3.16 - 0.980i)T \)
good3 \( 1 + (-0.907 + 0.589i)T + (1.22 - 2.74i)T^{2} \)
13 \( 1 + (-0.567 + 3.58i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-3.15 - 1.21i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (-2.20 + 2.44i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (2.16 - 8.06i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.72 + 0.884i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.70 + 0.914i)T + (30.3 - 6.44i)T^{2} \)
37 \( 1 + (-5.68 + 8.75i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (-4.04 + 1.31i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.101 - 0.101i)T + 43iT^{2} \)
47 \( 1 + (0.391 - 0.0205i)T + (46.7 - 4.91i)T^{2} \)
53 \( 1 + (-3.33 + 4.11i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (-7.54 - 8.38i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-13.4 - 1.41i)T + (59.6 + 12.6i)T^{2} \)
67 \( 1 + (-3.26 - 12.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.56 + 1.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.60 + 0.450i)T + (72.6 + 7.63i)T^{2} \)
79 \( 1 + (-6.72 + 15.0i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (2.68 - 0.425i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (1.64 + 2.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.14 + 0.498i)T + (92.2 + 29.9i)T^{2} \)
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

−10.06252670707333540541884322682, −9.246227481555137526661104047366, −8.615294874371403722374718692370, −7.87078118629598502376645799778, −7.34467966652298492677393157981, −5.86546009365178136068930109023, −5.25737963312308126420182649660, −3.76783430537698714755589581501, −2.51820191290423914263864635057, −1.17480600999881853357482919109, 0.932972973435568747611438041926, 2.76215271484278331467152459522, 3.71625495750629902288254228333, 4.29827772144118444535340725116, 6.39780273873257910153765757946, 6.78397838348037339911080235972, 7.950785805156850273009187005020, 8.497978173671584341521832060707, 9.651914069396443244986763252185, 10.03182603122774690336693664196

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