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Properties

Label 2-1470-7.2-c1-0-27
Degree $2$
Conductor $1470$
Sign $-0.198 - 0.980i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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 + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (−1.12 − 1.94i)11-s + (−0.499 + 0.866i)12-s − 5.65·13-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (1.29 + 2.23i)17-s + (0.499 + 0.866i)18-s + (−3.41 + 5.91i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.338 − 0.585i)11-s + (−0.144 + 0.249i)12-s − 1.56·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.313 + 0.543i)17-s + (0.117 + 0.204i)18-s + (−0.783 + 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1360585729\)
\(L(\frac12)\) \(\approx\) \(0.1360585729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (1.12 + 1.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + (-1.29 - 2.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.58 - 2.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
31 \( 1 + (5.12 + 8.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 + (1.12 - 1.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.171 + 0.297i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.58 + 2.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.82 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.53 + 7.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 + (0.414 - 0.717i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.48T + 97T^{2} \)
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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

−9.033871695600862103171163198207, −8.056329859591122473328506434657, −7.43393982840216195883424241741, −6.14313798403273699580511563347, −5.65358014830804173748002557410, −4.72008486033184417423451014318, −3.72875511584719350090703913168, −2.49691060280459913465394747565, −1.60082510913582060157244411076, −0.04638404322425592442024090621, 2.31754491779946910386437532338, 3.22056472245393277098161556369, 4.67733644652828494121895271079, 4.86578942203608537805531286060, 5.94587283220095251474719299979, 7.01579982547907727536048720917, 7.26013168043114568770945737659, 8.511516220089825590815907148828, 9.282623450521942835936122393577, 10.08730009244067626904883177209

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