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Properties

Label 2-70-5.3-c4-0-6
Degree $2$
Conductor $70$
Sign $-0.839 + 0.543i$
Analytic cond. $7.23589$
Root an. cond. $2.68996$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)2-s + (−6.14 − 6.14i)3-s − 8i·4-s + (22.5 + 10.7i)5-s + 24.5·6-s + (−13.0 + 13.0i)7-s + (16 + 16i)8-s − 5.51i·9-s + (−66.6 + 23.7i)10-s − 107.·11-s + (−49.1 + 49.1i)12-s + (−156. − 156. i)13-s − 52.3i·14-s + (−72.9 − 204. i)15-s − 64·16-s + (−345. + 345. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.682 − 0.682i)3-s − 0.5i·4-s + (0.903 + 0.428i)5-s + 0.682·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s − 0.0680i·9-s + (−0.666 + 0.237i)10-s − 0.890·11-s + (−0.341 + 0.341i)12-s + (−0.925 − 0.925i)13-s − 0.267i·14-s + (−0.324 − 0.909i)15-s − 0.250·16-s + (−1.19 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.839 + 0.543i$
Analytic conductor: \(7.23589\)
Root analytic conductor: \(2.68996\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :2),\ -0.839 + 0.543i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0774999 - 0.262365i\)
\(L(\frac12)\) \(\approx\) \(0.0774999 - 0.262365i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 2i)T \)
5 \( 1 + (-22.5 - 10.7i)T \)
7 \( 1 + (13.0 - 13.0i)T \)
good3 \( 1 + (6.14 + 6.14i)T + 81iT^{2} \)
11 \( 1 + 107.T + 1.46e4T^{2} \)
13 \( 1 + (156. + 156. i)T + 2.85e4iT^{2} \)
17 \( 1 + (345. - 345. i)T - 8.35e4iT^{2} \)
19 \( 1 + 553. iT - 1.30e5T^{2} \)
23 \( 1 + (313. + 313. i)T + 2.79e5iT^{2} \)
29 \( 1 + 154. iT - 7.07e5T^{2} \)
31 \( 1 + 1.05e3T + 9.23e5T^{2} \)
37 \( 1 + (1.00e3 - 1.00e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 2.59e3T + 2.82e6T^{2} \)
43 \( 1 + (826. + 826. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.70e3 + 1.70e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-770. - 770. i)T + 7.89e6iT^{2} \)
59 \( 1 + 5.56e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.89e3T + 1.38e7T^{2} \)
67 \( 1 + (3.50e3 - 3.50e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.07e3T + 2.54e7T^{2} \)
73 \( 1 + (2.47e3 + 2.47e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.24e3iT - 3.89e7T^{2} \)
83 \( 1 + (-572. - 572. i)T + 4.74e7iT^{2} \)
89 \( 1 - 3.33e3iT - 6.27e7T^{2} \)
97 \( 1 + (-6.88e3 + 6.88e3i)T - 8.85e7iT^{2} \)
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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

−13.30936336488436513141682567282, −12.64818371547406034484456793219, −11.04460288656488307623855703229, −10.13198072307693544607748250066, −8.843235191567507888503817765465, −7.27704327665132987478462916627, −6.33182676333986585245134659216, −5.32198022721434669893847125547, −2.32198798284744836096373328125, −0.16969571902749248531081170533, 2.17448586377371939542833986133, 4.44874171640991238672854631357, 5.67201308828527869760871509046, 7.42399348864257728998630734168, 9.150444330447002869410221912135, 9.963339042409253474014000267184, 10.81231052800754082439435268797, 12.00702212244843701776958144634, 13.17759801207655001202999999771, 14.20815036398290255083460850735

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