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Properties

Label 2-5070-13.12-c1-0-66
Degree $2$
Conductor $5070$
Sign $-0.691 + 0.722i$
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 − 1.35i·7-s + i·8-s + 9-s − 10-s − 3.19i·11-s + 12-s − 1.35·14-s + i·15-s + 16-s + 2.04·17-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 − 0.512i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.964i·11-s + 0.288·12-s − 0.362·14-s + 0.258i·15-s + 0.250·16-s + 0.496·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.691 + 0.722i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.603976804\)
\(L(\frac12)\) \(\approx\) \(1.603976804\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.35iT - 7T^{2} \)
11 \( 1 + 3.19iT - 11T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
19 \( 1 + 5.34iT - 19T^{2} \)
23 \( 1 - 8.32T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 + 0.185iT - 31T^{2} \)
37 \( 1 - 2.46iT - 37T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 2.91T + 43T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 - 5.74T + 53T^{2} \)
59 \( 1 - 1.62iT - 59T^{2} \)
61 \( 1 - 5.28T + 61T^{2} \)
67 \( 1 - 1.55iT - 67T^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 - 5.11iT - 73T^{2} \)
79 \( 1 + 1.42T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 3.37iT - 89T^{2} \)
97 \( 1 + 4.74iT - 97T^{2} \)
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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

−8.176857685145308346210784499726, −7.13467314997478405515940390645, −6.56874886863863816810710716006, −5.55684126284170420093762051828, −4.96429630381651544201914169697, −4.31049750589098281216614589243, −3.32869962824674179506218151530, −2.61244253344135603191622391879, −1.12064687734971819372143746988, −0.67699168484727859841905461558, 1.00434870795215488568264849062, 2.24504212541803203646051151749, 3.29711475573471965272622547439, 4.26540727912016278546452516353, 5.00586386337691403088419560195, 5.69912907947969463471966322563, 6.29556373872398008481587689932, 7.15004563762755285066587859670, 7.47145852076375266823940343542, 8.433546993558595963182479100832

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