Properties

Label 2-1520-380.379-c1-0-54
Degree $2$
Conductor $1520$
Sign $-0.974 + 0.223i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  + (−0.5 − 2.17i)5-s − 3·7-s + 3·9-s − 4.35i·11-s + 4.35i·17-s − 4.35i·19-s − 4·23-s + (−4.50 + 2.17i)25-s + (1.5 + 6.53i)35-s + 43-s + (−1.5 − 6.53i)45-s − 13·47-s + 2·49-s + (−9.50 + 2.17i)55-s − 15·61-s + ⋯
L(s)  = 1  + (−0.223 − 0.974i)5-s − 1.13·7-s + 9-s − 1.31i·11-s + 1.05i·17-s − 0.999i·19-s − 0.834·23-s + (−0.900 + 0.435i)25-s + (0.253 + 1.10i)35-s + 0.152·43-s + (−0.223 − 0.974i)45-s − 1.89·47-s + 0.285·49-s + (−1.28 + 0.293i)55-s − 1.92·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.974 + 0.223i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.974 + 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6711353768\)
\(L(\frac12)\) \(\approx\) \(0.6711353768\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.17i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 4.35iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.35iT - 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 13T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.103799287178670491724369060948, −8.382939353457827107705217795116, −7.59596642192616466280743094962, −6.49422194744953233670538523205, −5.95551863384690287101142259616, −4.83249911014058155297738801759, −3.94930041692814403084306975469, −3.14814889550664083957276970171, −1.57140202612128573553902789592, −0.25981117244050307337377077184, 1.81071199655440048009678262106, 2.93465351046656711873632424033, 3.83773599499473503083648746231, 4.67806006199149807135392483161, 5.98266824490886582893746105780, 6.76825466348911960981890906694, 7.26413045999532728984837754399, 7.995338139421489789655875907210, 9.407916954350084663762479816758, 9.943609902467584176150811570075

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