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Properties

Label 2-770-385.54-c1-0-38
Degree $2$
Conductor $770$
Sign $-0.656 + 0.754i$
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 + (−0.0752 − 0.130i)3-s + (−0.499 − 0.866i)4-s + (0.125 − 2.23i)5-s + 0.150·6-s + (−2.53 − 0.742i)7-s + 0.999·8-s + (1.48 − 2.57i)9-s + (1.87 + 1.22i)10-s + (0.899 + 3.19i)11-s + (−0.0752 + 0.130i)12-s − 2.51i·13-s + (1.91 − 1.82i)14-s + (−0.300 + 0.151i)15-s + (−0.5 + 0.866i)16-s + (−1.99 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.0434 − 0.0752i)3-s + (−0.249 − 0.433i)4-s + (0.0560 − 0.998i)5-s + 0.0614·6-s + (−0.959 − 0.280i)7-s + 0.353·8-s + (0.496 − 0.859i)9-s + (0.591 + 0.387i)10-s + (0.271 + 0.962i)11-s + (−0.0217 + 0.0376i)12-s − 0.697i·13-s + (0.511 − 0.488i)14-s + (−0.0775 + 0.0391i)15-s + (−0.125 + 0.216i)16-s + (−0.482 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.656 + 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204060 - 0.448153i\)
\(L(\frac12)\) \(\approx\) \(0.204060 - 0.448153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.125 + 2.23i)T \)
7 \( 1 + (2.53 + 0.742i)T \)
11 \( 1 + (-0.899 - 3.19i)T \)
good3 \( 1 + (0.0752 + 0.130i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 2.51iT - 13T^{2} \)
17 \( 1 + (1.99 - 1.14i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.16 - 5.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.94 + 2.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.85iT - 29T^{2} \)
31 \( 1 + (-1.82 + 1.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.79 + 5.65i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.710T + 41T^{2} \)
43 \( 1 + 2.99T + 43T^{2} \)
47 \( 1 + (3.34 - 5.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.78 - 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.75 + 4.47i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.84 + 3.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + (-11.5 + 6.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.29 + 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.40iT - 83T^{2} \)
89 \( 1 + (-3.02 - 1.74i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−9.932150731220213012155542730149, −9.171657201426608498548824053862, −8.302187839741585565911721517779, −7.44557407450755439863097086621, −6.40521877218353801249919286556, −5.89049988274812043838613434015, −4.48817933632950138181613025220, −3.80689758878331769585317038447, −1.81215112622178280299233696086, −0.26993836326328218010719445068, 1.97934566282434066585356825155, 3.00689402126005217405926583168, 3.90875113996489279701501069764, 5.21864582628560906855125824913, 6.58821615204932430405331552912, 6.95714559932410802188644542854, 8.268414342878840303575146568947, 9.071888541656507242959068279037, 9.936076378588273356557032945205, 10.60516061710722417356673823178

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