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Properties

Label 2-70-35.13-c1-0-1
Degree $2$
Conductor $70$
Sign $0.979 + 0.201i$
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 + (−0.941 + 2.47i)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 + (−0.355 + 0.934i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.979 + 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866432 - 0.0882676i\)
\(L(\frac12)\) \(\approx\) \(0.866432 - 0.0882676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.941 - 2.47i)T \)
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

−14.59657449981060710725166800223, −13.73460963481235310312476185764, −12.58552378802120746593658974618, −11.63020594915862570810532593404, −9.303791439939596796170436166651, −9.124475990955019392277946217372, −7.75192276755752972122424539104, −6.56757197148260786786250918424, −4.88199365542020945463572457055, −2.11189885120549725243992584967, 2.93893812303338789770631387506, 4.07963943454371272144407811349, 6.65806377191328104629730671593, 7.921038775700212950159562894875, 9.409940639487214685885581009183, 10.14868686409161879514923571805, 10.99134125831819050218991468451, 12.50552335021490797127893041958, 13.85977008777389613366498635225, 14.78470987378120487362941040263

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