Properties

Label 2-385-7.2-c1-0-14
Degree $2$
Conductor $385$
Sign $0.917 + 0.396i$
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  + (−0.981 + 1.69i)2-s + (−0.263 − 0.456i)3-s + (−0.926 − 1.60i)4-s + (0.5 − 0.866i)5-s + 1.03·6-s + (−2.64 + 0.138i)7-s − 0.289·8-s + (1.36 − 2.35i)9-s + (0.981 + 1.69i)10-s + (0.5 + 0.866i)11-s + (−0.488 + 0.845i)12-s + 0.196·13-s + (2.35 − 4.62i)14-s − 0.527·15-s + (2.13 − 3.70i)16-s + (−2.50 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.693 + 1.20i)2-s + (−0.152 − 0.263i)3-s + (−0.463 − 0.802i)4-s + (0.223 − 0.387i)5-s + 0.422·6-s + (−0.998 + 0.0522i)7-s − 0.102·8-s + (0.453 − 0.785i)9-s + (0.310 + 0.537i)10-s + (0.150 + 0.261i)11-s + (−0.140 + 0.244i)12-s + 0.0545·13-s + (0.630 − 1.23i)14-s − 0.136·15-s + (0.534 − 0.925i)16-s + (−0.607 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.917 + 0.396i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.917 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649642 - 0.134389i\)
\(L(\frac12)\) \(\approx\) \(0.649642 - 0.134389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.64 - 0.138i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.981 - 1.69i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.263 + 0.456i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 0.196T + 13T^{2} \)
17 \( 1 + (2.50 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.88 + 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.54 + 6.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.443T + 29T^{2} \)
31 \( 1 + (-1.93 - 3.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.11 + 8.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.83T + 41T^{2} \)
43 \( 1 - 0.847T + 43T^{2} \)
47 \( 1 + (6.58 - 11.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.556 - 0.963i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.74 + 9.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.45 + 5.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.44 - 7.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.95T + 71T^{2} \)
73 \( 1 + (1.29 + 2.23i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.77 - 4.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + (-1.95 + 3.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.41T + 97T^{2} \)
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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

−11.24574736424686773460821938228, −9.759048178694505560575688147672, −9.354842432513605232480645194588, −8.567532575743734403901692582299, −7.20641589961906968740382813070, −6.76841850222991737251013952066, −5.93274309733919216684568071485, −4.64957048806416531804118131087, −2.95043118897201081985579006653, −0.57748178529690345077042541449, 1.61767780093480005346365146231, 2.97445826197895092424510395039, 3.95561036470048804163024370757, 5.63568254719500729691108860060, 6.64601026908708447191904865310, 7.944359103281020496077888544719, 9.033237754532933682973276427547, 9.959668324477978242952111535956, 10.29160647202718475303363526158, 11.22345593861506871080296722772

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