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Properties

Label 2-770-11.4-c1-0-8
Degree $2$
Conductor $770$
Sign $-0.327 - 0.944i$
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 + (0.0759 − 0.233i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.198 − 0.144i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (2.37 + 1.72i)9-s − 10-s + (−3.31 − 0.0151i)11-s + 0.245·12-s + (0.931 + 0.677i)13-s + (−0.309 + 0.951i)14-s + (0.0759 + 0.233i)15-s + (−0.809 + 0.587i)16-s + (−5.30 + 3.85i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.0438 − 0.135i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (0.0812 − 0.0590i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.792 + 0.575i)9-s − 0.316·10-s + (−0.999 − 0.00455i)11-s + 0.0709·12-s + (0.258 + 0.187i)13-s + (−0.0825 + 0.254i)14-s + (0.0196 + 0.0603i)15-s + (−0.202 + 0.146i)16-s + (−1.28 + 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.327 - 0.944i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05251 + 1.47831i\)
\(L(\frac12)\) \(\approx\) \(1.05251 + 1.47831i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.0151i)T \)
good3 \( 1 + (-0.0759 + 0.233i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-0.931 - 0.677i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.30 - 3.85i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.06 - 3.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.30T + 23T^{2} \)
29 \( 1 + (0.906 + 2.78i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.12 - 3.72i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.19 + 3.67i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.26 - 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.95T + 43T^{2} \)
47 \( 1 + (1.90 - 5.86i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.33 + 3.14i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.692 + 2.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.07 + 5.86i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + (-0.307 + 0.223i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.23 + 6.87i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.83 - 1.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.16 + 6.65i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.48T + 89T^{2} \)
97 \( 1 + (2.79 + 2.03i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.81167709322031839708193193793, −9.792079197714026963311164513782, −8.480701359867835003666314062474, −8.018677368509736524127576967197, −6.98947860189747642694395441997, −6.30114011757043934045788751924, −5.07676048907761696072150507045, −4.39597970265109203792801735129, −3.15139869836321740933095976900, −1.94910235144953301933116409020, 0.76652484081228053478780132746, 2.46777985945349701545628240215, 3.58307159195698252326036693221, 4.64305661873777476680571008978, 5.17503759034735192230726495027, 6.71429128162278588664725466582, 7.19354648176130101803349872700, 8.493332199677135224440234124336, 9.287467526764209845589366598954, 10.24546179464376258524112166889

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