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Properties

Label 2-70-35.17-c1-0-2
Degree $2$
Conductor $70$
Sign $0.514 + 0.857i$
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  + (0.258 − 0.965i)2-s + (−0.279 + 0.0749i)3-s + (−0.866 − 0.499i)4-s + (0.774 − 2.09i)5-s + 0.289i·6-s + (2.64 + 0.126i)7-s + (−0.707 + 0.707i)8-s + (−2.52 + 1.45i)9-s + (−1.82 − 1.29i)10-s + (−2.81 + 4.87i)11-s + (0.279 + 0.0749i)12-s + (1.42 + 1.42i)13-s + (0.806 − 2.51i)14-s + (−0.0593 + 0.645i)15-s + (0.500 + 0.866i)16-s + (−1.37 − 5.12i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.161 + 0.0432i)3-s + (−0.433 − 0.249i)4-s + (0.346 − 0.938i)5-s + 0.118i·6-s + (0.998 + 0.0477i)7-s + (−0.249 + 0.249i)8-s + (−0.841 + 0.486i)9-s + (−0.577 − 0.408i)10-s + (−0.848 + 1.46i)11-s + (0.0807 + 0.0216i)12-s + (0.396 + 0.396i)13-s + (0.215 − 0.673i)14-s + (−0.0153 + 0.166i)15-s + (0.125 + 0.216i)16-s + (−0.333 − 1.24i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837558 - 0.474004i\)
\(L(\frac12)\) \(\approx\) \(0.837558 - 0.474004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.774 + 2.09i)T \)
7 \( 1 + (-2.64 - 0.126i)T \)
good3 \( 1 + (0.279 - 0.0749i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.81 - 4.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.42 - 1.42i)T + 13iT^{2} \)
17 \( 1 + (1.37 + 5.12i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.08 - 0.290i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.30 - 4.86i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \)
47 \( 1 + (5.69 + 1.52i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.357 + 1.33i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.73 - 4.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.816 - 0.218i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (-5.42 + 1.45i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.41 - 3.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.67 + 5.67i)T + 83iT^{2} \)
89 \( 1 + (-5.96 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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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

−14.23496870309034326148429216659, −13.41055863102942792547953909219, −12.20626191480806824928657945146, −11.39681389103787840318486118603, −10.12123429253051003017424655140, −8.943274005095355807520883954499, −7.72745330772669012513494326472, −5.42192922731815185982257089141, −4.64012837008708609477223636900, −2.08601485607357978177166730682, 3.23113492590834531484586472725, 5.39892185710597206040204015958, 6.32679529992169830999678336189, 7.87076587675380866114352400659, 8.837413198203463658440996534073, 10.74248657032693750467321161213, 11.29332290172481509905903858914, 13.04838728953603993673609460120, 14.07811231682168388820188551461, 14.76922015766755995503618618877

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