Properties

Label 2-1520-5.4-c1-0-31
Degree $2$
Conductor $1520$
Sign $0.871 - 0.490i$
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  + 2.31i·3-s + (1.94 − 1.09i)5-s − 1.45i·7-s − 2.35·9-s + 3.89·11-s − 3.05i·13-s + (2.53 + 4.50i)15-s + 3.92i·17-s − 19-s + 3.35·21-s − 5.37i·23-s + (2.59 − 4.27i)25-s + 1.48i·27-s − 6·29-s + 8.43·31-s + ⋯
L(s)  = 1  + 1.33i·3-s + (0.871 − 0.490i)5-s − 0.548i·7-s − 0.785·9-s + 1.17·11-s − 0.848i·13-s + (0.655 + 1.16i)15-s + 0.951i·17-s − 0.229·19-s + 0.732·21-s − 1.12i·23-s + (0.518 − 0.855i)25-s + 0.286i·27-s − 1.11·29-s + 1.51·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.180503795\)
\(L(\frac12)\) \(\approx\) \(2.180503795\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 + 1.09i)T \)
19 \( 1 + T \)
good3 \( 1 - 2.31iT - 3T^{2} \)
7 \( 1 + 1.45iT - 7T^{2} \)
11 \( 1 - 3.89T + 11T^{2} \)
13 \( 1 + 3.05iT - 13T^{2} \)
17 \( 1 - 3.92iT - 17T^{2} \)
23 \( 1 + 5.37iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8.43T + 31T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 1.45iT - 43T^{2} \)
47 \( 1 + 4.90iT - 47T^{2} \)
53 \( 1 - 4.23iT - 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 9.84iT - 67T^{2} \)
71 \( 1 + 8.64T + 71T^{2} \)
73 \( 1 - 2.43iT - 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 3.05iT - 97T^{2} \)
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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

−9.584272906234917930009041741558, −8.935092565829132543591981688809, −8.267653623230450589794011268556, −6.99702524159739055856326257125, −6.03432924167100060448243938987, −5.37947372048141449164175924831, −4.22835473865243625158924067621, −3.97095696805821747978407600552, −2.54144655553614049674004015769, −1.06177830867490609455736254626, 1.24430115390164282025223495926, 2.05211428901038910096461996650, 2.97729090037875183977493975615, 4.36707317979347761959484140263, 5.65987073874730840388547730354, 6.28750902976619387170876964867, 6.92895632433040077874757000796, 7.53162910719900506889360512450, 8.658124091843305838128789837718, 9.380992397046174638964661275772

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