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Properties

Label 2-70-7.4-c5-0-3
Degree $2$
Conductor $70$
Sign $-0.857 - 0.514i$
Analytic cond. $11.2268$
Root an. cond. $3.35065$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (2.5 − 4.33i)3-s + (−7.99 + 13.8i)4-s + (12.5 + 21.6i)5-s + 20·6-s + (−129.5 + 6.06i)7-s − 63.9·8-s + (109 + 188. i)9-s + (−50 + 86.6i)10-s + (−99 + 171. i)11-s + (40 + 69.2i)12-s − 340·13-s + (−280 − 436. i)14-s + 125·15-s + (−128 − 221. i)16-s + (−924 + 1.60e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.160 − 0.277i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.226·6-s + (−0.998 + 0.0467i)7-s − 0.353·8-s + (0.448 + 0.776i)9-s + (−0.158 + 0.273i)10-s + (−0.246 + 0.427i)11-s + (0.0801 + 0.138i)12-s − 0.557·13-s + (−0.381 − 0.595i)14-s + 0.143·15-s + (−0.125 − 0.216i)16-s + (−0.775 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(11.2268\)
Root analytic conductor: \(3.35065\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :5/2),\ -0.857 - 0.514i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.374593 + 1.35298i\)
\(L(\frac12)\) \(\approx\) \(0.374593 + 1.35298i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
5 \( 1 + (-12.5 - 21.6i)T \)
7 \( 1 + (129.5 - 6.06i)T \)
good3 \( 1 + (-2.5 + 4.33i)T + (-121.5 - 210. i)T^{2} \)
11 \( 1 + (99 - 171. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 340T + 3.71e5T^{2} \)
17 \( 1 + (924 - 1.60e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-605 - 1.04e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.41e3 + 2.44e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.53e3T + 2.05e7T^{2} \)
31 \( 1 + (-356 + 616. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.66e3 - 6.34e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.56e4T + 1.15e8T^{2} \)
43 \( 1 - 1.58e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.59e3 - 2.76e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.00e4 + 1.73e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.15e4 - 1.99e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-189.5 - 328. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.77e4 + 3.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 7.18e4T + 1.80e9T^{2} \)
73 \( 1 + (1.58e4 - 2.74e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.26e3 + 7.39e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.06e5T + 3.93e9T^{2} \)
89 \( 1 + (-6.15e3 - 1.06e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.02e5T + 8.58e9T^{2} \)
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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

−14.17704042735976303182475425158, −13.06810686539639181414109615066, −12.49681330125349014212948077723, −10.67202128760450160263158045824, −9.615981464939446532254673832769, −8.063212761338225666644413297938, −6.97808007619399794113824395566, −5.87098505741777862083072821391, −4.16776028052871354680035889944, −2.37299346199333441657401993383, 0.53016089204916867254959861934, 2.73566417908546534770182078952, 4.13332282596684449426055873799, 5.66011118624982935362372402259, 7.15308618117431266403498791125, 9.283146219143145701945513513468, 9.581185210997684631340416538409, 11.09017081842942578354861287459, 12.29687145835012708864666806683, 13.16997275657711140505202234691

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