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Properties

Label 2-770-7.2-c1-0-13
Degree $2$
Conductor $770$
Sign $0.0477 + 0.998i$
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 + (0.173 + 0.300i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.347·6-s + (−2.64 − 0.0412i)7-s − 0.999·8-s + (1.43 − 2.49i)9-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 − 0.300i)12-s + 6.45·13-s + (−1.35 + 2.27i)14-s − 0.347·15-s + (−0.5 + 0.866i)16-s + (−1.06 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.100 + 0.173i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.141·6-s + (−0.999 − 0.0155i)7-s − 0.353·8-s + (0.479 − 0.831i)9-s + (0.158 + 0.273i)10-s + (0.150 + 0.261i)11-s + (0.0501 − 0.0868i)12-s + 1.78·13-s + (−0.363 + 0.606i)14-s − 0.0896·15-s + (−0.125 + 0.216i)16-s + (−0.257 − 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0477 + 0.998i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0477 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19596 - 1.14014i\)
\(L(\frac12)\) \(\approx\) \(1.19596 - 1.14014i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0412i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.173 - 0.300i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 6.45T + 13T^{2} \)
17 \( 1 + (1.06 + 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.95 + 5.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.29 + 7.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.66T + 29T^{2} \)
31 \( 1 + (4.17 + 7.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.47 - 9.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (-4.92 + 8.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.70 - 2.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.75 - 3.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.35 - 5.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.911 - 1.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.22T + 71T^{2} \)
73 \( 1 + (-0.194 - 0.337i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.34 - 4.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 + (4.64 - 8.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.30T + 97T^{2} \)
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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

−10.22682586418936889919369126022, −9.178094121750270556898943429927, −8.941090021417643380330450295470, −7.25734281494469187973003990203, −6.57961660850672987238437509332, −5.70231735997795595700383575205, −4.24650830881553165874716667843, −3.60124487963091757082956220586, −2.64954393651518450803603802840, −0.823740010754640125040292392454, 1.51407498424481858843173342193, 3.44544647469250009323419495424, 3.93354500556420829386624152723, 5.49313962969131029322023422857, 5.93469281292653473084796512885, 7.18485887160393240285761929923, 7.72042180985744507176179445946, 8.856445096461925640149611681142, 9.329150621869144386683061199235, 10.65570749626563585495003221225

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