Properties

Label 2-770-385.152-c1-0-1
Degree $2$
Conductor $770$
Sign $0.0193 - 0.999i$
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  + (−0.0523 + 0.998i)2-s + (−1.17 − 3.06i)3-s + (−0.994 − 0.104i)4-s + (−2.19 + 0.444i)5-s + (3.12 − 1.01i)6-s + (0.953 − 2.46i)7-s + (0.156 − 0.987i)8-s + (−5.78 + 5.20i)9-s + (−0.328 − 2.21i)10-s + (−1.83 + 2.76i)11-s + (0.849 + 3.17i)12-s + (−2.05 + 1.04i)13-s + (2.41 + 1.08i)14-s + (3.93 + 6.19i)15-s + (0.978 + 0.207i)16-s + (−0.330 − 6.30i)17-s + ⋯
L(s)  = 1  + (−0.0370 + 0.706i)2-s + (−0.679 − 1.76i)3-s + (−0.497 − 0.0522i)4-s + (−0.980 + 0.198i)5-s + (1.27 − 0.414i)6-s + (0.360 − 0.932i)7-s + (0.0553 − 0.349i)8-s + (−1.92 + 1.73i)9-s + (−0.103 − 0.699i)10-s + (−0.552 + 0.833i)11-s + (0.245 + 0.915i)12-s + (−0.570 + 0.290i)13-s + (0.645 + 0.289i)14-s + (1.01 + 1.59i)15-s + (0.244 + 0.0519i)16-s + (−0.0802 − 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0193 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200576 + 0.196725i\)
\(L(\frac12)\) \(\approx\) \(0.200576 + 0.196725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0523 - 0.998i)T \)
5 \( 1 + (2.19 - 0.444i)T \)
7 \( 1 + (-0.953 + 2.46i)T \)
11 \( 1 + (1.83 - 2.76i)T \)
good3 \( 1 + (1.17 + 3.06i)T + (-2.22 + 2.00i)T^{2} \)
13 \( 1 + (2.05 - 1.04i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.330 + 6.30i)T + (-16.9 + 1.77i)T^{2} \)
19 \( 1 + (0.0151 + 0.144i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-1.07 - 4.02i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.567 - 0.781i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.15 - 5.42i)T + (-28.3 + 12.6i)T^{2} \)
37 \( 1 + (-2.91 - 1.11i)T + (27.4 + 24.7i)T^{2} \)
41 \( 1 + (-5.43 - 7.47i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.476 + 0.476i)T - 43iT^{2} \)
47 \( 1 + (2.28 + 1.85i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (6.57 - 10.1i)T + (-21.5 - 48.4i)T^{2} \)
59 \( 1 + (0.401 - 3.81i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-1.66 + 7.81i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (2.42 - 9.03i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.19 + 3.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.13 + 6.58i)T + (15.1 - 71.4i)T^{2} \)
79 \( 1 + (0.745 - 0.671i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (7.30 - 14.3i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (8.79 - 15.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.19 + 14.1i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.85241991163433107011892841572, −9.576323300300264401439577061102, −8.205941866306837776789393497713, −7.56831223586295957713547062817, −7.19419210606830969251866181223, −6.62017081268832319853029564924, −5.24455684591822294697395826794, −4.56529891306226384281299628338, −2.78557618522884216498765468150, −1.14985390492074994409829342450, 0.18234168789166719903288117065, 2.74424885148810425420929443889, 3.75393289832445877797679981441, 4.50744446430526341429365879391, 5.34335402204016459450752301258, 6.09329309679786321716740353191, 8.064776145792770416677059305089, 8.601114308528862251411819502533, 9.411264174373929321920911825810, 10.34150490175814969717545061636

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