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Properties

Label 2-2380-85.84-c1-0-47
Degree $2$
Conductor $2380$
Sign $0.709 + 0.704i$
Analytic cond. $19.0043$
Root an. cond. $4.35940$
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  + 3.16·3-s + (−0.0171 − 2.23i)5-s + 7-s + 6.98·9-s − 2.27i·11-s − 1.38i·13-s + (−0.0543 − 7.06i)15-s + (−2.92 + 2.90i)17-s + 6.80·19-s + 3.16·21-s + 3.58·23-s + (−4.99 + 0.0768i)25-s + 12.6·27-s + 9.18i·29-s + 6.18i·31-s + ⋯
L(s)  = 1  + 1.82·3-s + (−0.00768 − 0.999i)5-s + 0.377·7-s + 2.32·9-s − 0.685i·11-s − 0.383i·13-s + (−0.0140 − 1.82i)15-s + (−0.709 + 0.704i)17-s + 1.56·19-s + 0.689·21-s + 0.748·23-s + (−0.999 + 0.0153i)25-s + 2.42·27-s + 1.70i·29-s + 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(19.0043\)
Root analytic conductor: \(4.35940\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.833850201\)
\(L(\frac12)\) \(\approx\) \(3.833850201\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.0171 + 2.23i)T \)
7 \( 1 - T \)
17 \( 1 + (2.92 - 2.90i)T \)
good3 \( 1 - 3.16T + 3T^{2} \)
11 \( 1 + 2.27iT - 11T^{2} \)
13 \( 1 + 1.38iT - 13T^{2} \)
19 \( 1 - 6.80T + 19T^{2} \)
23 \( 1 - 3.58T + 23T^{2} \)
29 \( 1 - 9.18iT - 29T^{2} \)
31 \( 1 - 6.18iT - 31T^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + 9.12iT - 43T^{2} \)
47 \( 1 + 2.98iT - 47T^{2} \)
53 \( 1 + 0.939iT - 53T^{2} \)
59 \( 1 + 1.35T + 59T^{2} \)
61 \( 1 + 2.45iT - 61T^{2} \)
67 \( 1 + 4.66iT - 67T^{2} \)
71 \( 1 + 4.19iT - 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 2.49iT - 83T^{2} \)
89 \( 1 - 4.18T + 89T^{2} \)
97 \( 1 + 6.50T + 97T^{2} \)
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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

−8.784184491275957627594138569910, −8.445750983768608210812982028940, −7.51309800861057386590290040960, −6.95908149261277111498406839783, −5.46128077778768398239547428723, −4.85766548714969669972077817524, −3.67989947488483337596403169579, −3.25841417891197253432324532754, −2.00299360135010452154003823846, −1.16192033516430025963079976738, 1.57452597491718194959758797246, 2.56262481044086142710299903759, 3.03372989182924626786223109513, 4.07369939336778342588545882282, 4.76336030638374365650802534559, 6.19739119969916255616721072816, 7.21237159880005066799294594396, 7.48441972755269945164321908658, 8.205898989803696472283664479539, 9.169113444740872832844626624535

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