Properties

Label 2-1170-13.10-c1-0-6
Degree $2$
Conductor $1170$
Sign $0.702 - 0.711i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (2.59 + 1.5i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−0.232 + 0.133i)11-s + (0.866 − 3.5i)13-s − 3·14-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + (4.96 + 2.86i)19-s + (0.866 + 0.499i)20-s + (0.133 − 0.232i)22-s + (1.73 + 3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (0.981 + 0.566i)7-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (−0.0699 + 0.0403i)11-s + (0.240 − 0.970i)13-s − 0.801·14-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + (1.13 + 0.657i)19-s + (0.193 + 0.111i)20-s + (0.0285 − 0.0494i)22-s + (0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420830827\)
\(L(\frac12)\) \(\approx\) \(1.420830827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (-0.866 + 3.5i)T \)
good7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.232 - 0.133i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.732 - 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.92iT - 31T^{2} \)
37 \( 1 + (5.13 - 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.46 + 2i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.46iT - 47T^{2} \)
53 \( 1 - 0.267T + 53T^{2} \)
59 \( 1 + (-9.92 - 5.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.267 + 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 0.732i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.1 - 6.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 - 9.46iT - 83T^{2} \)
89 \( 1 + (-12.2 + 7.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.26 - 4.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.862896833096185143461726991375, −9.026709687332962814081391364541, −8.050525675847891896833512009260, −7.67503664069155845111921605553, −6.71098240277930954339833224241, −5.48215738354254951062616620048, −5.22028760132195090630312365516, −3.56620769204012685251646693123, −2.46473093580853033710750156575, −1.13197727366347250905797577908, 0.989877565214531816736208764081, 1.95699256349154985658221363673, 3.42349608106645822069929463647, 4.43760280876518864695344612559, 5.26013810330017264984537899787, 6.54924859775625570488700779780, 7.40216830499976534252252097908, 8.167411013090395168725179062426, 8.858667869542109779709892581906, 9.609141117060972563105060950752

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