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Properties

Label 2-42e2-9.4-c1-0-11
Degree $2$
Conductor $1764$
Sign $-0.954 - 0.299i$
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  + (0.899 + 1.47i)3-s + (−1.19 + 2.06i)5-s + (−1.38 + 2.66i)9-s + (1.12 + 1.95i)11-s + (−2.37 + 4.12i)13-s + (−4.12 + 0.0937i)15-s + 4.30·17-s + 8.59·19-s + (−0.664 + 1.15i)23-s + (−0.343 − 0.595i)25-s + (−5.18 + 0.353i)27-s + (−3.87 − 6.71i)29-s + (0.405 − 0.702i)31-s + (−1.87 + 3.43i)33-s − 4.63·37-s + ⋯
L(s)  = 1  + (0.519 + 0.854i)3-s + (−0.533 + 0.923i)5-s + (−0.460 + 0.887i)9-s + (0.340 + 0.590i)11-s + (−0.659 + 1.14i)13-s + (−1.06 + 0.0242i)15-s + 1.04·17-s + 1.97·19-s + (−0.138 + 0.240i)23-s + (−0.0687 − 0.119i)25-s + (−0.997 + 0.0680i)27-s + (−0.720 − 1.24i)29-s + (0.0727 − 0.126i)31-s + (−0.327 + 0.597i)33-s − 0.761·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.954 - 0.299i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.591067210\)
\(L(\frac12)\) \(\approx\) \(1.591067210\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.899 - 1.47i)T \)
7 \( 1 \)
good5 \( 1 + (1.19 - 2.06i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.12 - 1.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.37 - 4.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.30T + 17T^{2} \)
19 \( 1 - 8.59T + 19T^{2} \)
23 \( 1 + (0.664 - 1.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.87 + 6.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.405 + 0.702i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.63T + 37T^{2} \)
41 \( 1 + (5.00 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.74 + 3.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.18 + 3.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (2.40 - 4.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.575 + 0.997i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.06 + 3.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (-4.23 - 7.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.817 + 1.41i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.34T + 89T^{2} \)
97 \( 1 + (5.98 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.787779796843341147099755689853, −9.106232426065079840212520447728, −7.86632038881775686583076533848, −7.49384265291209321831410650814, −6.62886235806004223610277976367, −5.41547438286584084589029695745, −4.62375272758219644344984676403, −3.62505145891025749528231019118, −3.08874631378687095862182860803, −1.85395951234835193805083400945, 0.57864575001140377987201137754, 1.48339491347311379881444950711, 3.08751551629647152099020237609, 3.52805109951419290894252523989, 5.06628715720092684326323710880, 5.54341221172753516372684457339, 6.72071651821924502786381314161, 7.71419774647337601822774990662, 7.912794711489195619391823134720, 8.867147486039285350804636708741

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