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Properties

Label 2-70-35.13-c1-0-3
Degree $2$
Conductor $70$
Sign $0.709 + 0.704i$
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 + (0.541 − 0.541i)3-s − 1.00i·4-s + (−2.23 − 0.158i)5-s − 0.765i·6-s + (1.55 + 2.14i)7-s + (−0.707 − 0.707i)8-s + 2.41i·9-s + (−1.68 + 1.46i)10-s − 2.82·11-s + (−0.541 − 0.541i)12-s + (2.83 − 2.83i)13-s + (2.61 + 0.414i)14-s + (−1.29 + 1.12i)15-s − 1.00·16-s + (1.53 + 1.53i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.312 − 0.312i)3-s − 0.500i·4-s + (−0.997 − 0.0708i)5-s − 0.312i·6-s + (0.587 + 0.809i)7-s + (−0.250 − 0.250i)8-s + 0.804i·9-s + (−0.534 + 0.463i)10-s − 0.852·11-s + (−0.156 − 0.156i)12-s + (0.786 − 0.786i)13-s + (0.698 + 0.110i)14-s + (−0.333 + 0.289i)15-s − 0.250·16-s + (0.371 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.709 + 0.704i$
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: \((2,\ 70,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03517 - 0.426560i\)
\(L(\frac12)\) \(\approx\) \(1.03517 - 0.426560i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.158i)T \)
7 \( 1 + (-1.55 - 2.14i)T \)
good3 \( 1 + (-0.541 + 0.541i)T - 3iT^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (-2.83 + 2.83i)T - 13iT^{2} \)
17 \( 1 + (-1.53 - 1.53i)T + 17iT^{2} \)
19 \( 1 + 7.07T + 19T^{2} \)
23 \( 1 + (2.41 + 2.41i)T + 23iT^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 + 3.69iT - 31T^{2} \)
37 \( 1 + (-5.41 + 5.41i)T - 37iT^{2} \)
41 \( 1 - 1.53iT - 41T^{2} \)
43 \( 1 + (-4 - 4i)T + 43iT^{2} \)
47 \( 1 + (2.61 + 2.61i)T + 47iT^{2} \)
53 \( 1 + (-0.242 - 0.242i)T + 53iT^{2} \)
59 \( 1 - 3.82T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + (6.48 - 6.48i)T - 67iT^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + (-4.77 + 4.77i)T - 73iT^{2} \)
79 \( 1 - 9.07iT - 79T^{2} \)
83 \( 1 + (5.45 - 5.45i)T - 83iT^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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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

−14.65623325463360571351479656474, −13.21628724775994049998882943255, −12.55333840121385503185430017922, −11.29851779630329402381334620667, −10.51185031710145582993958243683, −8.487602544523923664017689457570, −7.85927429299248757757125324790, −5.80269736865371601942855919682, −4.34714517064839696655961222057, −2.49398922054417643061211859424, 3.58589607845968787871267683122, 4.61186103028457701914632051657, 6.57503830327895080777047985474, 7.79600326359745513209330029133, 8.788634649514958751010422261998, 10.55491785123610102053552139526, 11.61222589253796268710097186622, 12.78911009055174607590716488114, 14.04186313654186145371724770179, 14.86495883359545340948585141707

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