Properties

Label 2-70-7.2-c1-0-0
Degree $2$
Conductor $70$
Sign $0.266 - 0.963i$
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  + (−0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 1.99·6-s + (−2 + 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−1.5 − 2.59i)11-s + (0.999 − 1.73i)12-s + 5·13-s + (−0.499 − 2.59i)14-s + 1.99·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 0.816·6-s + (−0.755 + 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.452 − 0.783i)11-s + (0.288 − 0.499i)12-s + 1.38·13-s + (−0.133 − 0.694i)14-s + 0.516·15-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699032 + 0.531792i\)
\(L(\frac12)\) \(\approx\) \(0.699032 + 0.531792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−15.46884703286770629763712996237, −13.95269645933841741104857906200, −13.16981319807470047694460110414, −11.39122308004680081391904868716, −10.01262366676259156385878379647, −9.115226895573535331806530616824, −8.432608378056397329194640698368, −6.49009573887889420405119395625, −5.15545847896666123364690517603, −3.36443215474449269685995637012, 1.99133160960014135620033600093, 3.75548285742172031241020376086, 6.39989130578970503796910265395, 7.53085383801062044797924814498, 8.669190217470503777638136271838, 10.09854726735841160559946814581, 10.98886607033032661351288790856, 12.69204090063733291776161049702, 13.15044261811706142703383224756, 14.07200851584952180984754432976

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