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Properties

Label 2-1470-35.27-c1-0-21
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 + 1.51·11-s + (−0.707 + 0.707i)12-s + (3.93 + 3.93i)13-s + (−0.707 + 2.12i)15-s − 1.00·16-s + (3.07 − 3.07i)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 + 0.457·11-s + (−0.204 + 0.204i)12-s + (1.09 + 1.09i)13-s + (−0.182 + 0.547i)15-s − 0.250·16-s + (0.745 − 0.745i)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

\(L(1)\) \(\approx\) \(2.870761425\)
\(L(\frac12)\) \(\approx\) \(2.870761425\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + (-3.93 - 3.93i)T + 13iT^{2} \)
17 \( 1 + (-3.07 + 3.07i)T - 17iT^{2} \)
19 \( 1 + 0.585T + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 - 6.68iT - 41T^{2} \)
43 \( 1 + (3.83 - 3.83i)T - 43iT^{2} \)
47 \( 1 + (-3.97 + 3.97i)T - 47iT^{2} \)
53 \( 1 + (7.02 - 7.02i)T - 53iT^{2} \)
59 \( 1 + 0.729T + 59T^{2} \)
61 \( 1 - 3.03iT - 61T^{2} \)
67 \( 1 + (9.93 + 9.93i)T + 67iT^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 + (6.48 + 6.48i)T + 73iT^{2} \)
79 \( 1 + 6.68iT - 79T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 - 18.0T + 89T^{2} \)
97 \( 1 + (-12.2 + 12.2i)T - 97iT^{2} \)
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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

−9.575860929664408935548134662568, −9.149243683409927620091104014311, −7.968820766592933730034672611873, −7.38764486783953618285673247265, −6.23381404293601079498494999538, −6.00953805086212946015911181427, −4.63473372965886071550517012679, −3.82839669801905200317932739456, −3.01312690194867594595298593356, −1.83915058182690162626550193474, 0.997133763620858756906723820775, 1.82431936289482297556513046435, 3.17387871627074608708787713535, 3.92980982918934743297182900184, 5.07616576630788134177926625251, 5.84000386740564623725081425403, 6.55239890711671862805448491157, 7.79878235602988873229016095113, 8.620014715792036000097429385396, 9.037155232890630026168147976400

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