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Properties

Label 2-70-35.27-c1-0-3
Degree $2$
Conductor $70$
Sign $-0.501 + 0.865i$
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  + (−0.707 − 0.707i)2-s + (−1.30 − 1.30i)3-s + 1.00i·4-s + (−0.158 − 2.23i)5-s + 1.84i·6-s + (−2.47 − 0.941i)7-s + (0.707 − 0.707i)8-s + 0.414i·9-s + (−1.46 + 1.68i)10-s + 2.82·11-s + (1.30 − 1.30i)12-s + (4.23 + 4.23i)13-s + (1.08 + 2.41i)14-s + (−2.70 + 3.12i)15-s − 1.00·16-s + (3.69 − 3.69i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.754 − 0.754i)3-s + 0.500i·4-s + (−0.0708 − 0.997i)5-s + 0.754i·6-s + (−0.934 − 0.355i)7-s + (0.250 − 0.250i)8-s + 0.138i·9-s + (−0.463 + 0.534i)10-s + 0.852·11-s + (0.377 − 0.377i)12-s + (1.17 + 1.17i)13-s + (0.289 + 0.645i)14-s + (−0.698 + 0.805i)15-s − 0.250·16-s + (0.896 − 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ -0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275934 - 0.478596i\)
\(L(\frac12)\) \(\approx\) \(0.275934 - 0.478596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.158 + 2.23i)T \)
7 \( 1 + (2.47 + 0.941i)T \)
good3 \( 1 + (1.30 + 1.30i)T + 3iT^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + (-4.23 - 4.23i)T + 13iT^{2} \)
17 \( 1 + (-3.69 + 3.69i)T - 17iT^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + (-0.414 + 0.414i)T - 23iT^{2} \)
29 \( 1 + 0.828iT - 29T^{2} \)
31 \( 1 + 1.53iT - 31T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 + 3.69iT - 41T^{2} \)
43 \( 1 + (-4 + 4i)T - 43iT^{2} \)
47 \( 1 + (1.08 - 1.08i)T - 47iT^{2} \)
53 \( 1 + (8.24 - 8.24i)T - 53iT^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 - 6.43iT - 61T^{2} \)
67 \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (4.14 + 4.14i)T + 73iT^{2} \)
79 \( 1 - 5.07iT - 79T^{2} \)
83 \( 1 + (5.31 + 5.31i)T + 83iT^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-4.59 + 4.59i)T - 97iT^{2} \)
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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

−13.91077868627615360816419126902, −12.93229717701271587717069925202, −12.06961382274605391263286556825, −11.34388916344984759154183866504, −9.675937977332983795818312033210, −8.818977164039660260582417989386, −7.15435746260915370456338921797, −6.04506168858075457189373479891, −3.96264512066317996909061166558, −1.09516995505325869631902976096, 3.58870751755677526487013057970, 5.72631884677360801430512556421, 6.47683998663292806679556344576, 8.122609584170909833823622036421, 9.673867697241607160000685179460, 10.49339236670784065097735587354, 11.33451841861808858502740160951, 12.88395634919079196835124913755, 14.38826966323137517897453471004, 15.38830254736359799067700772880

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