Properties

Label 2-5070-13.12-c1-0-58
Degree $2$
Conductor $5070$
Sign $0.832 + 0.554i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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 − 3-s − 4-s i·5-s i·6-s − 2i·7-s i·8-s + 9-s + 10-s − 5i·11-s + 12-s + 2·14-s + i·15-s + 16-s + 2·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.50i·11-s + 0.288·12-s + 0.534·14-s + 0.258i·15-s + 0.250·16-s + 0.485·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431196738\)
\(L(\frac12)\) \(\approx\) \(1.431196738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 11iT - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 15iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.120528454565688642950939325201, −7.43305691806305787979894907511, −6.60284507937044046917999370680, −6.05743375122740498678500479902, −5.32222379829402757503548059111, −4.68482692401855903216629956485, −3.79976857185222834521731438940, −3.05037079576461632985435591999, −1.30012115050580688730725716993, −0.58381853848495046382908327236, 0.909375857703112440824229733450, 2.24870790747808629568340072805, 2.57310560744625012538482499110, 3.95172934677347510245855299813, 4.44352471960726017744804309082, 5.46792732898408456677176856269, 5.87222355214376018319351739419, 7.04596303977487781270613693294, 7.36379449332286867902451332506, 8.442056504694076169330668583022

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