Properties

Label 2-1470-21.20-c1-0-22
Degree $2$
Conductor $1470$
Sign $-0.0595 - 0.998i$
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 + (1.53 − 0.801i)3-s − 4-s + 5-s + (0.801 + 1.53i)6-s i·8-s + (1.71 − 2.46i)9-s + i·10-s + 5.48i·11-s + (−1.53 + 0.801i)12-s + 4.37i·13-s + (1.53 − 0.801i)15-s + 16-s − 6.99·17-s + (2.46 + 1.71i)18-s + 4.64i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.886 − 0.462i)3-s − 0.5·4-s + 0.447·5-s + (0.327 + 0.626i)6-s − 0.353i·8-s + (0.571 − 0.820i)9-s + 0.316i·10-s + 1.65i·11-s + (−0.443 + 0.231i)12-s + 1.21i·13-s + (0.396 − 0.206i)15-s + 0.250·16-s − 1.69·17-s + (0.580 + 0.404i)18-s + 1.06i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0595 - 0.998i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.0595 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.148264640\)
\(L(\frac12)\) \(\approx\) \(2.148264640\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 + 0.801i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 5.48iT - 11T^{2} \)
13 \( 1 - 4.37iT - 13T^{2} \)
17 \( 1 + 6.99T + 17T^{2} \)
19 \( 1 - 4.64iT - 19T^{2} \)
23 \( 1 - 7.99iT - 23T^{2} \)
29 \( 1 + 3.40iT - 29T^{2} \)
31 \( 1 + 3.85iT - 31T^{2} \)
37 \( 1 - 9.25T + 37T^{2} \)
41 \( 1 - 4.36T + 41T^{2} \)
43 \( 1 - 1.48T + 43T^{2} \)
47 \( 1 - 7.51T + 47T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 - 6.00T + 59T^{2} \)
61 \( 1 + 6.75iT - 61T^{2} \)
67 \( 1 + 5.98T + 67T^{2} \)
71 \( 1 - 4.42iT - 71T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 - 4.87T + 79T^{2} \)
83 \( 1 + 9.22T + 83T^{2} \)
89 \( 1 - 0.525T + 89T^{2} \)
97 \( 1 + 2.60iT - 97T^{2} \)
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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

−9.412192726566890319990085342312, −9.056018731365537684009890362912, −7.899414342983958987839710605974, −7.34000382765846318673982251267, −6.64188095259051111544403891797, −5.86756468359003077826778109884, −4.46188324127981005005991485452, −4.03952224889516527981499961281, −2.38824110342141800498314224383, −1.66808566339502291478152375419, 0.77269980381626976427152649982, 2.51561231604682575882969705502, 2.86179318235264413439659400827, 4.03241290222529713338877252498, 4.87409574626638398736598530349, 5.85832517628970742836149987912, 6.90017313197991719605495560367, 8.143917112006342602542648108611, 8.725677067879512040890674577102, 9.133035471954188492552589357866

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