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Properties

Label 2-770-7.4-c1-0-13
Degree $2$
Conductor $770$
Sign $0.722 + 0.690i$
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 + (0.267 − 0.462i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.534·6-s + (1.70 − 2.02i)7-s + 0.999·8-s + (1.35 + 2.35i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.267 + 0.462i)12-s − 1.91·13-s + (−2.60 − 0.462i)14-s + 0.534·15-s + (−0.5 − 0.866i)16-s + (1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.154 − 0.267i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.218·6-s + (0.643 − 0.765i)7-s + 0.353·8-s + (0.452 + 0.783i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (0.0771 + 0.133i)12-s − 0.529·13-s + (−0.696 − 0.123i)14-s + 0.137·15-s + (−0.125 − 0.216i)16-s + (0.348 − 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.722 + 0.690i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.722 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43662 - 0.576076i\)
\(L(\frac12)\) \(\approx\) \(1.43662 - 0.576076i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.70 + 2.02i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.267 + 0.462i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 1.91T + 13T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.99 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.33 - 2.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.33T + 29T^{2} \)
31 \( 1 + (0.590 - 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.68 + 8.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (1.40 + 2.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.96 + 6.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.36 - 5.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.32 + 2.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.03 - 6.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 + (-0.793 + 1.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.50 + 9.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.68T + 83T^{2} \)
89 \( 1 + (3.40 + 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.96T + 97T^{2} \)
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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

−10.31530572702879084378477746945, −9.582023049458386081886956197264, −8.487272692653872732636334543910, −7.47213394206599963241614090056, −7.28235662998139173750792384437, −5.67334125575497538058436893359, −4.64045576902416596597529279892, −3.55382619675883082612742018506, −2.31863934878555695647393335900, −1.19189807358548497888906531441, 1.19254810134096219167214871596, 2.73022227300074643461848316501, 4.33376934552919594886453232436, 5.05710742240311926761691842049, 6.05771819166250197613010707284, 6.98190943263102315584518548731, 7.932097664135947938587342643767, 8.857687357763723345836491520351, 9.337407113288473446842415950908, 10.13384738408045354984661225371

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