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Properties

Label 2-770-385.164-c1-0-4
Degree $2$
Conductor $770$
Sign $-0.387 - 0.921i$
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  + (−0.5 − 0.866i)2-s + (−1.64 + 2.85i)3-s + (−0.499 + 0.866i)4-s + (−0.0163 − 2.23i)5-s + 3.29·6-s + (−1.01 − 2.44i)7-s + 0.999·8-s + (−3.93 − 6.81i)9-s + (−1.92 + 1.13i)10-s + (2.63 + 2.01i)11-s + (−1.64 − 2.85i)12-s + 3.27i·13-s + (−1.60 + 2.10i)14-s + (6.40 + 3.63i)15-s + (−0.5 − 0.866i)16-s + (1.23 + 0.711i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.951 + 1.64i)3-s + (−0.249 + 0.433i)4-s + (−0.00731 − 0.999i)5-s + 1.34·6-s + (−0.383 − 0.923i)7-s + 0.353·8-s + (−1.31 − 2.27i)9-s + (−0.609 + 0.358i)10-s + (0.794 + 0.607i)11-s + (−0.475 − 0.824i)12-s + 0.908i·13-s + (−0.430 + 0.561i)14-s + (1.65 + 0.939i)15-s + (−0.125 − 0.216i)16-s + (0.298 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.387 - 0.921i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262588 + 0.395164i\)
\(L(\frac12)\) \(\approx\) \(0.262588 + 0.395164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.0163 + 2.23i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
11 \( 1 + (-2.63 - 2.01i)T \)
good3 \( 1 + (1.64 - 2.85i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 3.27iT - 13T^{2} \)
17 \( 1 + (-1.23 - 0.711i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.91 - 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.95 - 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.06iT - 29T^{2} \)
31 \( 1 + (4.72 + 2.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.46 - 3.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.52T + 41T^{2} \)
43 \( 1 - 3.88T + 43T^{2} \)
47 \( 1 + (-6.42 - 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.28 + 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.91 - 5.72i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.842 - 1.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.99 + 2.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.01T + 71T^{2} \)
73 \( 1 + (-11.7 - 6.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.362 - 0.209i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + (7.47 - 4.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.51T + 97T^{2} \)
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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

−10.46660294637976624765118933402, −9.616119657254092518278469665541, −9.494740346743286218046384102015, −8.456295716474985524308971639500, −7.06488780353250109987437763531, −5.91464460313987036566493452511, −4.90075351538772219595305081399, −4.08460997774335777014880355011, −3.65647604233638941165356864881, −1.29805161531868450648838925901, 0.33491865379764108668014854471, 1.95040080170136499591398523853, 3.17356804102983214128014314438, 5.38327882887173687194907433980, 5.83561430466284836366204444153, 6.69094210383587543160593867192, 7.15504956584329920400994928285, 8.089466212417295101239723775709, 8.908247034486281164465942786291, 10.18591763635717224209012656925

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