Properties

Label 2-5070-13.12-c1-0-93
Degree $2$
Conductor $5070$
Sign $-0.999 - 0.0304i$
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 − 4.80i·7-s + i·8-s + 9-s − 10-s − 2.55i·11-s − 12-s − 4.80·14-s i·15-s + 16-s + 3.02·17-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 − 1.81i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.770i·11-s − 0.288·12-s − 1.28·14-s − 0.258i·15-s + 0.250·16-s + 0.734·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.999 - 0.0304i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.999 - 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.234388232\)
\(L(\frac12)\) \(\approx\) \(2.234388232\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.80iT - 7T^{2} \)
11 \( 1 + 2.55iT - 11T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 - 5.44iT - 31T^{2} \)
37 \( 1 + 7.51iT - 37T^{2} \)
41 \( 1 + 5.89iT - 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 6.42iT - 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 + 7.60iT - 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 0.929iT - 67T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 + 5.40T + 79T^{2} \)
83 \( 1 - 4.33iT - 83T^{2} \)
89 \( 1 - 16.8iT - 89T^{2} \)
97 \( 1 - 17.9iT - 97T^{2} \)
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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

−7.971982403271934515787959402319, −7.27617785855618452015490769567, −6.63709283889020826384918550694, −5.43629867612921401370716544008, −4.69873405430538934203655818283, −3.88680264408018960750061816419, −3.44032922867463791936741232613, −2.44782463179688390390343418303, −1.17181104551611772204828101335, −0.62061807626077117394380634180, 1.52404693768390337656619097040, 2.56835891833709634066077370664, 3.12969651618580874200166212718, 4.27688525027894986172771287928, 5.03192701170317853579303953829, 5.91887619298224809967256700326, 6.27443176037765733499698891669, 7.28126795518163592444351890915, 7.930868801918087454452386864180, 8.448320642899004409904954787448

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