Properties

Label 2-770-385.157-c1-0-45
Degree $2$
Conductor $770$
Sign $-0.625 + 0.780i$
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  + (−0.358 − 0.933i)2-s + (0.420 + 0.647i)3-s + (−0.743 + 0.669i)4-s + (1.98 − 1.03i)5-s + (0.453 − 0.624i)6-s + (0.552 − 2.58i)7-s + (0.891 + 0.453i)8-s + (0.977 − 2.19i)9-s + (−1.67 − 1.48i)10-s + (−3.05 − 1.28i)11-s + (−0.745 − 0.199i)12-s + (−6.49 − 1.02i)13-s + (−2.61 + 0.411i)14-s + (1.50 + 0.849i)15-s + (0.104 − 0.994i)16-s + (1.03 − 2.69i)17-s + ⋯
L(s)  = 1  + (−0.253 − 0.660i)2-s + (0.242 + 0.373i)3-s + (−0.371 + 0.334i)4-s + (0.886 − 0.461i)5-s + (0.185 − 0.254i)6-s + (0.208 − 0.977i)7-s + (0.315 + 0.160i)8-s + (0.325 − 0.732i)9-s + (−0.529 − 0.468i)10-s + (−0.921 − 0.387i)11-s + (−0.215 − 0.0576i)12-s + (−1.80 − 0.285i)13-s + (−0.698 + 0.110i)14-s + (0.387 + 0.219i)15-s + (0.0261 − 0.248i)16-s + (0.251 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563758 - 1.17449i\)
\(L(\frac12)\) \(\approx\) \(0.563758 - 1.17449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.358 + 0.933i)T \)
5 \( 1 + (-1.98 + 1.03i)T \)
7 \( 1 + (-0.552 + 2.58i)T \)
11 \( 1 + (3.05 + 1.28i)T \)
good3 \( 1 + (-0.420 - 0.647i)T + (-1.22 + 2.74i)T^{2} \)
13 \( 1 + (6.49 + 1.02i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.03 + 2.69i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (2.93 - 3.25i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (0.0198 + 0.00532i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.52 - 0.494i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.795 + 0.0836i)T + (30.3 - 6.44i)T^{2} \)
37 \( 1 + (-3.42 - 2.22i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (-1.90 + 0.618i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.26 - 5.26i)T - 43iT^{2} \)
47 \( 1 + (0.358 + 6.84i)T + (-46.7 + 4.91i)T^{2} \)
53 \( 1 + (3.22 + 2.61i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (-1.05 - 1.16i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-4.25 - 0.447i)T + (59.6 + 12.6i)T^{2} \)
67 \( 1 + (-15.1 + 4.05i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.15 - 3.74i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.748 + 14.2i)T + (-72.6 - 7.63i)T^{2} \)
79 \( 1 + (0.710 - 1.59i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-1.37 - 8.67i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.35 - 5.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.67 + 16.9i)T + (-92.2 - 29.9i)T^{2} \)
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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

−9.975027804364326986929852626573, −9.576247697583696968193190172060, −8.437023951319646210360442141010, −7.63905409041602688363124647117, −6.57648533022336792601550632521, −5.19133917607286305331561135200, −4.55132467820897167568052126481, −3.30317494618348840943661745842, −2.20453115485829543243934957943, −0.66723355963714661849519365619, 2.06746914085931673665287778697, 2.57158453979087365876882693597, 4.75639541978942137546720162385, 5.29444013996453606435790959848, 6.36599301580078684708013918254, 7.23310203632032896437616888536, 7.920997401606304758393942540137, 8.840291522738533933155430678431, 9.786914846896770637386719682929, 10.28360705925735057474065477992

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