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Properties

Label 2-770-7.4-c1-0-18
Degree $2$
Conductor $770$
Sign $0.841 + 0.540i$
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 + (0.766 − 1.32i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.53·6-s + (1.28 − 2.31i)7-s − 0.999·8-s + (0.326 + 0.565i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.766 + 1.32i)12-s + 4.36·13-s + (2.64 − 0.0412i)14-s − 1.53·15-s + (−0.5 − 0.866i)16-s + (−2.17 + 3.76i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.442 − 0.766i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.625·6-s + (0.486 − 0.873i)7-s − 0.353·8-s + (0.108 + 0.188i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (0.221 + 0.383i)12-s + 1.21·13-s + (0.707 − 0.0110i)14-s − 0.395·15-s + (−0.125 − 0.216i)16-s + (−0.527 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.841 + 0.540i$
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: \((2,\ 770,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05517 - 0.603656i\)
\(L(\frac12)\) \(\approx\) \(2.05517 - 0.603656i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 2.31i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.766 + 1.32i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 4.36T + 13T^{2} \)
17 \( 1 + (2.17 - 3.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.35 + 5.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.57 + 6.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.39T + 29T^{2} \)
31 \( 1 + (-3.16 + 5.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.946 + 1.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.69T + 41T^{2} \)
43 \( 1 + 9.27T + 43T^{2} \)
47 \( 1 + (-6.24 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.11 - 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.69 - 4.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 - 5.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.72 - 8.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.18T + 71T^{2} \)
73 \( 1 + (-2.56 + 4.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.53 + 6.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (-7.21 - 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.93T + 97T^{2} \)
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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

−10.43025614596517248608298846706, −8.853280978337124885463351141061, −8.364542124178847745599297673619, −7.73310099330658921396353516362, −6.71673178489522859000101714237, −6.13557112690426985857947948086, −4.55398916188080026491709125293, −4.14523446000455361843267053214, −2.53362466259531859117986177573, −1.04275670089890098378935675216, 1.69076683723959721842828482822, 3.00779831493272973660905101418, 3.83974323224865568147710855787, 4.70250763735579129072389412007, 5.81943859998022051032326476831, 6.73635600780624595579948468706, 8.228962349970608703199526919317, 8.756900392990296107782896499173, 9.721759653986950862797307512267, 10.33925110770069958849184605714

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