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Properties

Label 2-770-11.5-c1-0-2
Degree $2$
Conductor $770$
Sign $0.872 + 0.489i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−1.76 − 1.28i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.674 + 2.07i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.545 + 1.68i)9-s − 0.999·10-s + (2.35 + 2.33i)11-s + 2.18·12-s + (0.782 + 2.40i)13-s + (0.809 + 0.587i)14-s + (−1.76 + 1.28i)15-s + (0.309 − 0.951i)16-s + (−1.77 + 5.45i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−1.01 − 0.740i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.275 + 0.847i)6-s + (−0.305 + 0.222i)7-s + (0.286 + 0.207i)8-s + (0.181 + 0.560i)9-s − 0.316·10-s + (0.711 + 0.703i)11-s + 0.630·12-s + (0.217 + 0.668i)13-s + (0.216 + 0.157i)14-s + (−0.456 + 0.331i)15-s + (0.0772 − 0.237i)16-s + (−0.430 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792933 - 0.207277i\)
\(L(\frac12)\) \(\approx\) \(0.792933 - 0.207277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.35 - 2.33i)T \)
good3 \( 1 + (1.76 + 1.28i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-0.782 - 2.40i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.77 - 5.45i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.81 - 1.31i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + (-6.97 + 5.06i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.568 - 1.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.74 + 2.72i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.53 - 6.20i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + (2.60 + 1.89i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.24 + 3.82i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.62 - 1.18i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.0828 - 0.255i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.86T + 67T^{2} \)
71 \( 1 + (2.95 - 9.08i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-9.24 + 6.71i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.63 + 11.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.27 - 6.98i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + (-5.10 - 15.6i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28955519565776983751735548759, −9.533638951229843670254393814948, −8.675502129845139757394531633821, −7.68745038110006925640787210870, −6.48691313321720019712767388277, −6.08121421905464629406595180955, −4.76359635720066151879720679375, −3.83267354343159153371057148604, −2.11294237574936178537463602773, −1.09737667956544745149517158187, 0.64042516928844075467022594169, 3.00894854158764046733356370579, 4.26823884657360480933737737806, 5.18184971398797244866869317563, 6.02141530040033574510172888633, 6.68155386805838123083068248378, 7.66888840212924369369118607767, 8.800653160261370797848475647021, 9.641273179339976176114074431987, 10.33097072221407414263984479913

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