↑

Properties

Label 2-770-35.27-c1-0-33
Degree $2$
Conductor $770$
Sign $-0.367 + 0.930i$
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 + (0.903 + 0.903i)3-s + 1.00i·4-s + (1.41 − 1.72i)5-s − 1.27i·6-s + (−2.26 + 1.36i)7-s + (0.707 − 0.707i)8-s − 1.36i·9-s + (−2.22 + 0.221i)10-s − 11-s + (−0.903 + 0.903i)12-s + (−4.51 − 4.51i)13-s + (2.56 + 0.638i)14-s + (2.84 − 0.283i)15-s − 1.00·16-s + (4.77 − 4.77i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.521 + 0.521i)3-s + 0.500i·4-s + (0.633 − 0.773i)5-s − 0.521i·6-s + (−0.856 + 0.515i)7-s + (0.250 − 0.250i)8-s − 0.455i·9-s + (−0.703 + 0.0700i)10-s − 0.301·11-s + (−0.260 + 0.260i)12-s + (−1.25 − 1.25i)13-s + (0.686 + 0.170i)14-s + (0.734 − 0.0730i)15-s − 0.250·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.367 + 0.930i$
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: \((2,\ 770,\ (\ :1/2),\ -0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594471 - 0.874117i\)
\(L(\frac12)\) \(\approx\) \(0.594471 - 0.874117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.72i)T \)
7 \( 1 + (2.26 - 1.36i)T \)
11 \( 1 + T \)
good3 \( 1 + (-0.903 - 0.903i)T + 3iT^{2} \)
13 \( 1 + (4.51 + 4.51i)T + 13iT^{2} \)
17 \( 1 + (-4.77 + 4.77i)T - 17iT^{2} \)
19 \( 1 + 6.06T + 19T^{2} \)
23 \( 1 + (-1.28 + 1.28i)T - 23iT^{2} \)
29 \( 1 + 7.55iT - 29T^{2} \)
31 \( 1 - 7.78iT - 31T^{2} \)
37 \( 1 + (-4.82 - 4.82i)T + 37iT^{2} \)
41 \( 1 - 5.27iT - 41T^{2} \)
43 \( 1 + (-6.22 + 6.22i)T - 43iT^{2} \)
47 \( 1 + (4.65 - 4.65i)T - 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 + 0.913iT - 61T^{2} \)
67 \( 1 + (8.57 + 8.57i)T + 67iT^{2} \)
71 \( 1 - 6.02T + 71T^{2} \)
73 \( 1 + (-2.79 - 2.79i)T + 73iT^{2} \)
79 \( 1 - 6.11iT - 79T^{2} \)
83 \( 1 + (5.04 + 5.04i)T + 83iT^{2} \)
89 \( 1 + 7.76T + 89T^{2} \)
97 \( 1 + (-6.01 + 6.01i)T - 97iT^{2} \)
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

−9.858593382264892448046537545126, −9.455238809405780630354942383558, −8.599293319447592200315663873840, −7.83816695674983637331495942768, −6.54272665260866095472356511258, −5.48029487173351951846175643785, −4.51841700190610019780242438993, −3.10586130999242058494055436267, −2.51423515099528925614994318382, −0.56389728750313753405160321816, 1.79935445268326671707170147522, 2.72039504669868302315626812853, 4.14805392052402614425546801746, 5.58490545363435246064685484179, 6.48969798198668815921713791918, 7.20995915314887974040341687048, 7.75939974698288406715934104671, 8.916795542811545732388335976304, 9.693721075254051957349403488694, 10.39017207367760332664527851944

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