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Properties

Label 2-770-11.3-c1-0-19
Degree $2$
Conductor $770$
Sign $-0.889 + 0.456i$
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 + (−0.632 − 1.94i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−1.65 − 1.20i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.962 + 0.699i)9-s + 10-s + (−2.40 − 2.28i)11-s − 2.04·12-s + (2.83 − 2.05i)13-s + (0.309 + 0.951i)14-s + (0.632 − 1.94i)15-s + (−0.809 − 0.587i)16-s + (−3.46 − 2.51i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.365 − 1.12i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (−0.676 − 0.491i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.320 + 0.233i)9-s + 0.316·10-s + (−0.725 − 0.688i)11-s − 0.590·12-s + (0.785 − 0.570i)13-s + (0.0825 + 0.254i)14-s + (0.163 − 0.502i)15-s + (−0.202 − 0.146i)16-s + (−0.840 − 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.889 + 0.456i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393585 - 1.62981i\)
\(L(\frac12)\) \(\approx\) \(0.393585 - 1.62981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.40 + 2.28i)T \)
good3 \( 1 + (0.632 + 1.94i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-2.83 + 2.05i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.46 + 2.51i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.37 + 4.22i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + (0.403 - 1.24i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.61 + 3.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.101 - 0.311i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.19 - 3.68i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.63T + 43T^{2} \)
47 \( 1 + (-3.77 - 11.6i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.883 - 0.641i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.10 + 9.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-10.5 - 7.64i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + (-4.75 - 3.45i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.29 + 7.07i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.403 + 0.293i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.2 + 7.41i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.39T + 89T^{2} \)
97 \( 1 + (-7.92 + 5.75i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19275133330181411145026658534, −9.124084473743831235099430111455, −8.133188327775963844719909432303, −7.13103726914863510985936954513, −6.23250624162787520010098964058, −5.76320027185991726080199306443, −4.55538422483018808013035624053, −3.05784260619120562732255987435, −2.20675441755822059681518491898, −0.72451594878703769888333127444, 2.06708483945170847464608280469, 3.77307517734924616366720410343, 4.32727370279731265108008093240, 5.23660697368177376262661834389, 6.07323120588374022336819480792, 7.01321903156918870228346928962, 8.162798716433252273440924331134, 9.000575626312685594841894794205, 10.09456020116343428523528092582, 10.47227661266322492860217938972

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