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Properties

Label 2-770-11.3-c1-0-11
Degree $2$
Conductor $770$
Sign $0.970 + 0.242i$
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  + (0.809 − 0.587i)2-s + (0.118 + 0.363i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.309 + 0.224i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (2.30 − 1.67i)9-s + 10-s + (0.309 + 3.30i)11-s + 0.381·12-s + (1.61 − 1.17i)13-s + (0.309 + 0.951i)14-s + (−0.118 + 0.363i)15-s + (−0.809 − 0.587i)16-s + (0.5 + 0.363i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0681 + 0.209i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.126 + 0.0916i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (0.769 − 0.559i)9-s + 0.316·10-s + (0.0931 + 0.995i)11-s + 0.110·12-s + (0.448 − 0.326i)13-s + (0.0825 + 0.254i)14-s + (−0.0304 + 0.0937i)15-s + (−0.202 − 0.146i)16-s + (0.121 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43275 - 0.299049i\)
\(L(\frac12)\) \(\approx\) \(2.43275 - 0.299049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.309 - 3.30i)T \)
good3 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.5 - 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0450 + 0.138i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-0.381 + 1.17i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.61 - 1.90i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.763 + 2.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.64 + 5.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 + (-1.47 - 4.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.61 + 1.90i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.66 - 5.11i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.0 + 7.33i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + (3 + 2.17i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.20 - 6.79i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.381 + 0.277i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.30 + 3.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.90T + 89T^{2} \)
97 \( 1 + (8.54 - 6.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.36855435159328207639127163433, −9.532495694380540801508201955353, −8.935484204793636984062716569742, −7.46000381324245316374530781466, −6.70953956149675691126402222282, −5.75174126007722852476292903827, −4.75652446944338137937298844416, −3.80178593189049742422345854352, −2.73416033341834546448367524828, −1.44238467357729267598038128890, 1.34295148034909589265862608145, 2.88977057114442357074772784731, 4.04168758151204427292026511147, 4.98741469343264639820943897600, 5.94803595470040168534811668855, 6.81409651013972598465210708964, 7.61901819416257215867582329130, 8.549793413387922640408088387593, 9.363289206654113824273809965966, 10.49676271377110920966926817466

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