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Properties

Label 2-270-5.2-c2-0-13
Degree $2$
Conductor $270$
Sign $0.163 + 0.986i$
Analytic cond. $7.35696$
Root an. cond. $2.71237$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (−4.62 − 1.89i)5-s + (−6.13 − 6.13i)7-s + (−2 + 2i)8-s + (−2.73 − 6.52i)10-s + 8.32·11-s + (14.2 − 14.2i)13-s − 12.2i·14-s − 4·16-s + (−21.9 − 21.9i)17-s − 30.5i·19-s + (3.79 − 9.25i)20-s + (8.32 + 8.32i)22-s + (−17.1 + 17.1i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.925 − 0.379i)5-s + (−0.876 − 0.876i)7-s + (−0.250 + 0.250i)8-s + (−0.273 − 0.652i)10-s + 0.757·11-s + (1.09 − 1.09i)13-s − 0.876i·14-s − 0.250·16-s + (−1.29 − 1.29i)17-s − 1.61i·19-s + (0.189 − 0.462i)20-s + (0.378 + 0.378i)22-s + (−0.744 + 0.744i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.860870 - 0.729568i\)
\(L(\frac12)\) \(\approx\) \(0.860870 - 0.729568i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (4.62 + 1.89i)T \)
good7 \( 1 + (6.13 + 6.13i)T + 49iT^{2} \)
11 \( 1 - 8.32T + 121T^{2} \)
13 \( 1 + (-14.2 + 14.2i)T - 169iT^{2} \)
17 \( 1 + (21.9 + 21.9i)T + 289iT^{2} \)
19 \( 1 + 30.5iT - 361T^{2} \)
23 \( 1 + (17.1 - 17.1i)T - 529iT^{2} \)
29 \( 1 - 37.8iT - 841T^{2} \)
31 \( 1 + 13.8T + 961T^{2} \)
37 \( 1 + (27.6 + 27.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 19.1T + 1.68e3T^{2} \)
43 \( 1 + (-13.8 + 13.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (-3.65 - 3.65i)T + 2.20e3iT^{2} \)
53 \( 1 + (-13.0 + 13.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 18.2iT - 3.48e3T^{2} \)
61 \( 1 + 11.2T + 3.72e3T^{2} \)
67 \( 1 + (-79.5 - 79.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 15.3T + 5.04e3T^{2} \)
73 \( 1 + (-34.5 + 34.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 80.8iT - 6.24e3T^{2} \)
83 \( 1 + (-12.4 + 12.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 0.375iT - 7.92e3T^{2} \)
97 \( 1 + (37.9 + 37.9i)T + 9.40e3iT^{2} \)
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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

−11.49399276652225040518292736596, −10.83071477882756787958546343270, −9.318738735043460905227933682691, −8.547229381907031056131467167478, −7.22564927868087526383064206689, −6.76171757438334057167039629699, −5.26546511015823932722964130615, −4.07643847695168054145102059060, −3.25394227171755998090796040680, −0.48328298737525934096403058654, 1.99113156340892331599633803380, 3.62203056379191333528496468407, 4.18644401056445974935941764368, 6.18083600477869917402805286938, 6.46864443441938883095688482267, 8.245293521856778029784645794258, 9.035346082067799787486270679235, 10.21470343905843880546990182553, 11.18965799497690072790544971928, 11.98018112764859970659944272215

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