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Properties

Label 2-70-5.4-c1-0-0
Degree $2$
Conductor $70$
Sign $-0.100 - 0.994i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
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 + 2.44i·3-s − 4-s + (−0.224 − 2.22i)5-s − 2.44·6-s + i·7-s − i·8-s − 2.99·9-s + (2.22 − 0.224i)10-s + 4.89·11-s − 2.44i·12-s − 0.449i·13-s − 14-s + (5.44 − 0.550i)15-s + 16-s − 2i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (−0.100 − 0.994i)5-s − 0.999·6-s + 0.377i·7-s − 0.353i·8-s − 0.999·9-s + (0.703 − 0.0710i)10-s + 1.47·11-s − 0.707i·12-s − 0.124i·13-s − 0.267·14-s + (1.40 − 0.142i)15-s + 0.250·16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594372 + 0.657441i\)
\(L(\frac12)\) \(\approx\) \(0.594372 + 0.657441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (0.224 + 2.22i)T \)
7 \( 1 - iT \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 0.449iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 6.44T + 19T^{2} \)
23 \( 1 + 6.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + 0.898T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 8.89iT - 43T^{2} \)
47 \( 1 - 0.898iT - 47T^{2} \)
53 \( 1 + 1.10iT - 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.89iT - 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−15.08517839053954537072533383340, −14.42659898044210509523723155384, −12.88787510282672316243007210570, −11.70172461178631872541256264227, −10.20015222730524925012267609842, −9.099332830909959217552901262909, −8.513163680692904578369236277240, −6.41637525834994849944410655192, −4.92316433075312663206256433053, −4.04853735820217459631622764883, 1.82896760108220512789219906716, 3.74623558901870409569966711662, 6.31546491003053097024332362523, 7.16680802511913941244908265704, 8.555439577994667383484011730629, 10.14426114234254611513281680593, 11.38654572036831701617938777479, 12.12441150829848094183317813801, 13.30914428357588708925777978210, 14.08526841533397932505097033534

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