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Properties

Label 2-42e2-63.4-c1-0-13
Degree $2$
Conductor $1764$
Sign $-0.217 - 0.976i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.36 + 1.06i)3-s − 0.0223·5-s + (0.725 + 2.91i)9-s − 0.560·11-s + (−2.06 + 3.57i)13-s + (−0.0305 − 0.0238i)15-s + (1.29 − 2.23i)17-s + (1.69 + 2.93i)19-s + 4.49·23-s − 4.99·25-s + (−2.11 + 4.74i)27-s + (2.57 + 4.45i)29-s + (0.407 + 0.705i)31-s + (−0.765 − 0.597i)33-s + (−0.235 − 0.407i)37-s + ⋯
L(s)  = 1  + (0.787 + 0.615i)3-s − 0.0100·5-s + (0.241 + 0.970i)9-s − 0.169·11-s + (−0.572 + 0.991i)13-s + (−0.00788 − 0.00616i)15-s + (0.313 − 0.542i)17-s + (0.389 + 0.674i)19-s + 0.936·23-s − 0.999·25-s + (−0.407 + 0.913i)27-s + (0.477 + 0.827i)29-s + (0.0732 + 0.126i)31-s + (−0.133 − 0.104i)33-s + (−0.0387 − 0.0670i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.991409978\)
\(L(\frac12)\) \(\approx\) \(1.991409978\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.36 - 1.06i)T \)
7 \( 1 \)
good5 \( 1 + 0.0223T + 5T^{2} \)
11 \( 1 + 0.560T + 11T^{2} \)
13 \( 1 + (2.06 - 3.57i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.29 + 2.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.69 - 2.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.49T + 23T^{2} \)
29 \( 1 + (-2.57 - 4.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.407 - 0.705i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.235 + 0.407i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.02 - 5.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.811 - 1.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.05 - 10.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.45 + 2.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.47 - 7.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.54 + 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.16T + 71T^{2} \)
73 \( 1 + (-3.55 + 6.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.15 - 2.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.83 - 6.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.502 + 0.870i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.32 - 12.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.504659575896560489512349545795, −8.885464954880927813134322805339, −7.944906371408211326966599997465, −7.37007480792297828297300166169, −6.39290415180854150730851817276, −5.18232469350534754926915725432, −4.59721185472969836201157706563, −3.56195001098594091130711901825, −2.74529614060144213132415516628, −1.60285855439914431675894999687, 0.66975809740156988365075908369, 2.06386473604907057645517441468, 2.96785831133972853400155844695, 3.81500539396001527173484078619, 5.02585667820291656148882053231, 5.90544997252986168933476363267, 6.90347958786990857712793304117, 7.54815606990873267464745948211, 8.243522923669271984255809685158, 8.916162007901062884892072222424

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