Properties

Label 2-1470-21.20-c1-0-32
Degree $2$
Conductor $1470$
Sign $0.598 + 0.801i$
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  + i·2-s + (−1.24 − 1.20i)3-s − 4-s − 5-s + (1.20 − 1.24i)6-s i·8-s + (0.0774 + 2.99i)9-s i·10-s + 0.388i·11-s + (1.24 + 1.20i)12-s − 0.436i·13-s + (1.24 + 1.20i)15-s + 16-s − 0.581·17-s + (−2.99 + 0.0774i)18-s − 0.597i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.716 − 0.697i)3-s − 0.5·4-s − 0.447·5-s + (0.493 − 0.506i)6-s − 0.353i·8-s + (0.0258 + 0.999i)9-s − 0.316i·10-s + 0.117i·11-s + (0.358 + 0.348i)12-s − 0.120i·13-s + (0.320 + 0.312i)15-s + 0.250·16-s − 0.141·17-s + (−0.706 + 0.0182i)18-s − 0.137i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7691353345\)
\(L(\frac12)\) \(\approx\) \(0.7691353345\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.24 + 1.20i)T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 0.388iT - 11T^{2} \)
13 \( 1 + 0.436iT - 13T^{2} \)
17 \( 1 + 0.581T + 17T^{2} \)
19 \( 1 + 0.597iT - 19T^{2} \)
23 \( 1 - 7.83iT - 23T^{2} \)
29 \( 1 + 2.39iT - 29T^{2} \)
31 \( 1 + 8.53iT - 31T^{2} \)
37 \( 1 - 6.55T + 37T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 5.35T + 47T^{2} \)
53 \( 1 + 7.66iT - 53T^{2} \)
59 \( 1 - 5.33T + 59T^{2} \)
61 \( 1 + 14.8iT - 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 7.51iT - 73T^{2} \)
79 \( 1 - 4.78T + 79T^{2} \)
83 \( 1 + 0.157T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + 1.08iT - 97T^{2} \)
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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

−9.382643907100601244211702393025, −8.101225227362895940836246508574, −7.82056718149842108430156581819, −6.91466676648621742316438093891, −6.21948004813567767473444556738, −5.38818618978416037828661917195, −4.60347473645604474351040608636, −3.47186098467333788257091024825, −1.92786832399819344420416628726, −0.42333522564174161867751862787, 1.02395586436805386858193094452, 2.69521811536752727950274783659, 3.70188741566212413260141402117, 4.50804512044629942502295754370, 5.20319391530903141754774753099, 6.27762042275496526504014730024, 7.07110718708364990033454767018, 8.392928026910628163877089631317, 8.869218277319845605066896239242, 9.943189702017048296129546429639

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