Properties

Label 2-3850-5.4-c1-0-8
Degree $2$
Conductor $3850$
Sign $-0.447 + 0.894i$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
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  + i·2-s + 2.24i·3-s − 4-s − 2.24·6-s i·7-s i·8-s − 2.05·9-s + 11-s − 2.24i·12-s − 0.941i·13-s + 14-s + 16-s + 6.49i·17-s − 2.05i·18-s + 4.36·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.29i·3-s − 0.5·4-s − 0.918·6-s − 0.377i·7-s − 0.353i·8-s − 0.686·9-s + 0.301·11-s − 0.649i·12-s − 0.261i·13-s + 0.267·14-s + 0.250·16-s + 1.57i·17-s − 0.485i·18-s + 1.00·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3850} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8220491264\)
\(L(\frac12)\) \(\approx\) \(0.8220491264\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 - T \)
good3 \( 1 - 2.24iT - 3T^{2} \)
13 \( 1 + 0.941iT - 13T^{2} \)
17 \( 1 - 6.49iT - 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 + 6.24iT - 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 9.55T + 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 2.13T + 41T^{2} \)
43 \( 1 + 7.67iT - 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 - 4.74iT - 53T^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 - 9.11T + 61T^{2} \)
67 \( 1 - 12.9iT - 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 - 8.49iT - 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−8.911560705174966577768423786415, −8.407718185832291315719607728809, −7.45428529819656189325584654498, −6.84793502158442561027162365498, −5.71363236326418809815120718025, −5.43120000828547134664143974845, −4.19115501955114168834167316022, −4.04192966906447675163104374008, −3.06242875842848547306372795309, −1.48170825477880705552697959789, 0.23973666452012554217673179173, 1.45696956033664421330175220750, 2.06575998995198144726835608231, 3.09058305264751460760502114339, 3.86940011791107694387957253722, 5.18689702752912704862986819355, 5.59749153007878602046625024393, 6.69587090870748140447833636940, 7.44285010835918137853122438480, 7.69620767680516087844214756173

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