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Properties

Label 2-70-35.3-c1-0-2
Degree $2$
Conductor $70$
Sign $0.879 - 0.475i$
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  + (0.965 + 0.258i)2-s + (0.304 + 1.13i)3-s + (0.866 + 0.499i)4-s + (−1.79 − 1.33i)5-s + 1.17i·6-s + (−2.55 − 0.698i)7-s + (0.707 + 0.707i)8-s + (1.40 − 0.810i)9-s + (−1.38 − 1.75i)10-s + (−0.371 + 0.643i)11-s + (−0.304 + 1.13i)12-s + (2.05 − 2.05i)13-s + (−2.28 − 1.33i)14-s + (0.975 − 2.43i)15-s + (0.500 + 0.866i)16-s + (−6.33 + 1.69i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.175 + 0.655i)3-s + (0.433 + 0.249i)4-s + (−0.800 − 0.599i)5-s + 0.479i·6-s + (−0.964 − 0.264i)7-s + (0.249 + 0.249i)8-s + (0.467 − 0.270i)9-s + (−0.437 − 0.555i)10-s + (−0.112 + 0.194i)11-s + (−0.0877 + 0.327i)12-s + (0.570 − 0.570i)13-s + (−0.610 − 0.356i)14-s + (0.251 − 0.629i)15-s + (0.125 + 0.216i)16-s + (−1.53 + 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14392 + 0.289691i\)
\(L(\frac12)\) \(\approx\) \(1.14392 + 0.289691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (1.79 + 1.33i)T \)
7 \( 1 + (2.55 + 0.698i)T \)
good3 \( 1 + (-0.304 - 1.13i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.371 - 0.643i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.05 + 2.05i)T - 13iT^{2} \)
17 \( 1 + (6.33 - 1.69i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.946 - 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.36 - 5.11i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (-2.96 - 1.71i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.58 - 0.691i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 - 1.59i)T + 43iT^{2} \)
47 \( 1 + (-1.21 + 4.54i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.81 - 1.29i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.27 - 2.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.25 + 3.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.54 + 13.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + (2.29 + 8.54i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.70 - 3.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.23 + 9.23i)T - 83iT^{2} \)
89 \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \)
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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

−15.30144140036082863880022237972, −13.50315086705134504640918175855, −12.84328171793818964501641106795, −11.66969051927824308562629533076, −10.34130068210432360173203901630, −9.118065336764700610393717769184, −7.69910051178360014974579206056, −6.24950210949458876242615247001, −4.49211094209597580958041307485, −3.55515477497507570546307639620, 2.71472592698341343586250230479, 4.32003541561705982733983498162, 6.44774911070402593543710329884, 7.15247652275304834237301907098, 8.767547339068841993239339894565, 10.46510548982906028615607118371, 11.50403412980207724580130811846, 12.63607340346052709558085772073, 13.39076255442734390121486072430, 14.45031561816133273847672180862

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