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Properties

Label 2-1470-35.27-c1-0-7
Degree $2$
Conductor $1470$
Sign $-0.561 - 0.827i$
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 + (−1 − 2i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 2.12i)10-s − 4.34·11-s + (0.707 − 0.707i)12-s + (1.93 + 1.93i)13-s + (−0.707 + 2.12i)15-s − 1.00·16-s + (1.07 − 1.07i)17-s + (−0.707 + 0.707i)18-s + 0.585·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.447 − 0.894i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.223 − 0.670i)10-s − 1.30·11-s + (0.204 − 0.204i)12-s + (0.535 + 0.535i)13-s + (−0.182 + 0.547i)15-s − 0.250·16-s + (0.259 − 0.259i)17-s + (−0.166 + 0.166i)18-s + 0.134·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8209222912\)
\(L(\frac12)\) \(\approx\) \(0.8209222912\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
7 \( 1 \)
good11 \( 1 + 4.34T + 11T^{2} \)
13 \( 1 + (-1.93 - 1.93i)T + 13iT^{2} \)
17 \( 1 + (-1.07 + 1.07i)T - 17iT^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 - 9.90iT - 29T^{2} \)
31 \( 1 + 3.65iT - 31T^{2} \)
37 \( 1 + (-3.27 - 3.27i)T + 37iT^{2} \)
41 \( 1 - 5.03iT - 41T^{2} \)
43 \( 1 + (-6.17 + 6.17i)T - 43iT^{2} \)
47 \( 1 + (5.68 - 5.68i)T - 47iT^{2} \)
53 \( 1 + (9.45 - 9.45i)T - 53iT^{2} \)
59 \( 1 + 7.55T + 59T^{2} \)
61 \( 1 - 8.68iT - 61T^{2} \)
67 \( 1 + (-5.93 - 5.93i)T + 67iT^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (7.65 + 7.65i)T + 73iT^{2} \)
79 \( 1 - 5.03iT - 79T^{2} \)
83 \( 1 + (-1.61 - 1.61i)T + 83iT^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
97 \( 1 + (9.85 - 9.85i)T - 97iT^{2} \)
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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

−9.620240513818895120445080206786, −8.787636206888888761268874713434, −7.87774336328843225133417702345, −7.53550123617765426128471154240, −6.42358689946388166681792391649, −5.55261879840641306750055474742, −4.96095329000243408960033401030, −4.06665290248646968568967529796, −2.89118707973806293754929870590, −1.38834542533751330180072985237, 0.29503390752116171666283313696, 2.26195928426585544733270226910, 3.18648005958134462015518247487, 4.00355773232688221020966923605, 4.96874837640643033873518288610, 5.87761772599515754727256667003, 6.51062538809057198410008012831, 7.73199772900682438733639438634, 8.240671077356094461899546096025, 9.665538471448001918255728763074

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