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Properties

Label 2-385-11.3-c1-0-10
Degree $2$
Conductor $385$
Sign $0.881 + 0.472i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
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  + (−0.661 + 0.480i)2-s + (−0.404 − 1.24i)3-s + (−0.411 + 1.26i)4-s + (−0.809 − 0.587i)5-s + (0.865 + 0.628i)6-s + (−0.309 + 0.951i)7-s + (−0.841 − 2.58i)8-s + (1.04 − 0.757i)9-s + 0.817·10-s + (−2.33 + 2.35i)11-s + 1.74·12-s + (5.69 − 4.13i)13-s + (−0.252 − 0.777i)14-s + (−0.404 + 1.24i)15-s + (−0.354 − 0.257i)16-s + (4.58 + 3.32i)17-s + ⋯
L(s)  = 1  + (−0.467 + 0.339i)2-s + (−0.233 − 0.718i)3-s + (−0.205 + 0.633i)4-s + (−0.361 − 0.262i)5-s + (0.353 + 0.256i)6-s + (−0.116 + 0.359i)7-s + (−0.297 − 0.915i)8-s + (0.347 − 0.252i)9-s + 0.258·10-s + (−0.705 + 0.708i)11-s + 0.503·12-s + (1.57 − 1.14i)13-s + (−0.0674 − 0.207i)14-s + (−0.104 + 0.321i)15-s + (−0.0886 − 0.0644i)16-s + (1.11 + 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851788 - 0.214090i\)
\(L(\frac12)\) \(\approx\) \(0.851788 - 0.214090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (2.33 - 2.35i)T \)
good2 \( 1 + (0.661 - 0.480i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.404 + 1.24i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-5.69 + 4.13i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.58 - 3.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.21 + 6.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.87T + 23T^{2} \)
29 \( 1 + (0.733 - 2.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.80 + 4.94i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.72 + 8.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.545 - 1.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 + (-0.977 - 3.00i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.29 + 1.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.53 - 4.72i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.63 - 1.91i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.102T + 67T^{2} \)
71 \( 1 + (4.59 + 3.34i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.49 - 7.68i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.69 - 6.31i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.63 + 4.82i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.48T + 89T^{2} \)
97 \( 1 + (-9.09 + 6.60i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.35837533568005383485580695301, −10.29008605998765809230505503146, −9.161612902809423387582257710568, −8.313045359973678780110636888994, −7.62703144609169560469460492922, −6.76298084070782574650151995725, −5.69173362232152876144933243949, −4.22587848509962655932335871344, −2.97901886494887567109608141191, −0.865576893730550893571398753165, 1.31351248104011327546766856005, 3.28439487119465438545793279654, 4.45304201574203089336282247292, 5.51734168367378086717475696944, 6.54393102862164153294161763688, 7.964671330009654646861221551982, 8.790092419271443672865873517835, 9.892980157196279192321672124822, 10.42705224083406091998164989386, 11.11825534819932512062233376691

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