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Properties

Label 2-70-7.2-c3-0-5
Degree $2$
Conductor $70$
Sign $0.109 + 0.993i$
Analytic cond. $4.13013$
Root an. cond. $2.03227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (0.5 + 0.866i)3-s + (−1.99 − 3.46i)4-s + (2.5 − 4.33i)5-s + 1.99·6-s + (8.5 − 16.4i)7-s − 7.99·8-s + (13 − 22.5i)9-s + (−5 − 8.66i)10-s + (1 + 1.73i)11-s + (1.99 − 3.46i)12-s − 8·13-s + (−19.9 − 31.1i)14-s + 5·15-s + (−8 + 13.8i)16-s + (26 + 45.0i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0962 + 0.166i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.136·6-s + (0.458 − 0.888i)7-s − 0.353·8-s + (0.481 − 0.833i)9-s + (−0.158 − 0.273i)10-s + (0.0274 + 0.0474i)11-s + (0.0481 − 0.0833i)12-s − 0.170·13-s + (−0.381 − 0.595i)14-s + 0.0860·15-s + (−0.125 + 0.216i)16-s + (0.370 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.109 + 0.993i$
Analytic conductor: \(4.13013\)
Root analytic conductor: \(2.03227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :3/2),\ 0.109 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.36936 - 1.22626i\)
\(L(\frac12)\) \(\approx\) \(1.36936 - 1.22626i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-8.5 + 16.4i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 8T + 2.19e3T^{2} \)
17 \( 1 + (-26 - 45.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (13 - 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (33.5 - 58.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 69T + 2.43e4T^{2} \)
31 \( 1 + (-166 - 287. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (98 - 169. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 353T + 6.89e4T^{2} \)
43 \( 1 + 369T + 7.95e4T^{2} \)
47 \( 1 + (44 - 76.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (291 + 504. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-175 - 303. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-233.5 + 404. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (145.5 + 252. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 770T + 3.57e5T^{2} \)
73 \( 1 + (314 + 543. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (585 - 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 525T + 5.71e5T^{2} \)
89 \( 1 + (44.5 - 77.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−13.90017956161391562780667209272, −12.79087747875899032434507728206, −11.86904520697557713628736282217, −10.50432978411598396335334626351, −9.697309774893940696392087317012, −8.255800488813696362858871972577, −6.59437431472095699429908910563, −4.87418974724685251818133944659, −3.62896135393571611011497399557, −1.29954010762029141943497057740, 2.47755848146599701218814816680, 4.65860703636915936932312377835, 5.93891313420744606603028701489, 7.34819419645862299868845152426, 8.413840211524428609751120563425, 9.821725981542800403805216314986, 11.27960921055354730281736862891, 12.44463204221851683778371605495, 13.57252672117858168800399144401, 14.46198645234442961249175383338

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