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Properties

Label 2-1470-35.13-c1-0-18
Degree $2$
Conductor $1470$
Sign $0.566 + 0.824i$
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  + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−0.569 + 2.16i)5-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (1.12 + 1.93i)10-s − 2.01·11-s + (−0.707 − 0.707i)12-s + (3.44 − 3.44i)13-s + (1.12 + 1.93i)15-s − 1.00·16-s + (3.40 + 3.40i)17-s + (−0.707 − 0.707i)18-s + 6.72·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.254 + 0.966i)5-s − 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.356 + 0.610i)10-s − 0.607·11-s + (−0.204 − 0.204i)12-s + (0.954 − 0.954i)13-s + (0.290 + 0.498i)15-s − 0.250·16-s + (0.824 + 0.824i)17-s + (−0.166 − 0.166i)18-s + 1.54·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.566 + 0.824i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.560819979\)
\(L(\frac12)\) \(\approx\) \(2.560819979\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.569 - 2.16i)T \)
7 \( 1 \)
good11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 + (-3.44 + 3.44i)T - 13iT^{2} \)
17 \( 1 + (-3.40 - 3.40i)T + 17iT^{2} \)
19 \( 1 - 6.72T + 19T^{2} \)
23 \( 1 + (-4.65 - 4.65i)T + 23iT^{2} \)
29 \( 1 + 1.60iT - 29T^{2} \)
31 \( 1 + 6.25iT - 31T^{2} \)
37 \( 1 + (-3.63 + 3.63i)T - 37iT^{2} \)
41 \( 1 + 4.94iT - 41T^{2} \)
43 \( 1 + (4.68 + 4.68i)T + 43iT^{2} \)
47 \( 1 + (1.07 + 1.07i)T + 47iT^{2} \)
53 \( 1 + (-1.42 - 1.42i)T + 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 9.00iT - 61T^{2} \)
67 \( 1 + (1.19 - 1.19i)T - 67iT^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 + (-5.97 + 5.97i)T - 73iT^{2} \)
79 \( 1 - 13.9iT - 79T^{2} \)
83 \( 1 + (8.59 - 8.59i)T - 83iT^{2} \)
89 \( 1 - 2.40T + 89T^{2} \)
97 \( 1 + (-13.1 - 13.1i)T + 97iT^{2} \)
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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

−9.637941583523398149233672165985, −8.448126967310616644609545433165, −7.67970823009817018051282933701, −7.08734424030142232994662221191, −5.86748239533610004411432674689, −5.44811850078426686570098063400, −3.80856244597964739009220702146, −3.32227081024154629523415582018, −2.43705960127092727274448564975, −1.04310209863073920489897138161, 1.21595988038675845777326487127, 2.91136328930635370463895934832, 3.69924421119195664785603844940, 4.91394682550245085418666556702, 5.06509935523403313776685365120, 6.31189675464645731316448288335, 7.30514602714017926112875974396, 8.051611305300310347214899715820, 8.775926910776084425487752076013, 9.382664515039381557342395399866

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