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Properties

Label 2-42e2-9.4-c1-0-16
Degree $2$
Conductor $1764$
Sign $0.904 - 0.426i$
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  + (1.05 − 1.37i)3-s + (−0.736 + 1.27i)5-s + (−0.788 − 2.89i)9-s + (2.35 + 4.07i)11-s + (−1.23 + 2.13i)13-s + (0.981 + 2.35i)15-s + 2.88·17-s + 0.763·19-s + (1.12 − 1.94i)23-s + (1.41 + 2.44i)25-s + (−4.81 − 1.95i)27-s + (−0.583 − 1.01i)29-s + (−4.06 + 7.04i)31-s + (8.08 + 1.04i)33-s − 1.62·37-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)3-s + (−0.329 + 0.570i)5-s + (−0.262 − 0.964i)9-s + (0.709 + 1.22i)11-s + (−0.341 + 0.592i)13-s + (0.253 + 0.608i)15-s + 0.698·17-s + 0.175·19-s + (0.233 − 0.404i)23-s + (0.282 + 0.489i)25-s + (−0.926 − 0.376i)27-s + (−0.108 − 0.187i)29-s + (−0.730 + 1.26i)31-s + (1.40 + 0.182i)33-s − 0.266·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.904 - 0.426i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.972800100\)
\(L(\frac12)\) \(\approx\) \(1.972800100\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 + 1.37i)T \)
7 \( 1 \)
good5 \( 1 + (0.736 - 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 - 2.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 - 0.763T + 19T^{2} \)
23 \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.583 + 1.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.06 - 7.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.62T + 37T^{2} \)
41 \( 1 + (1.37 - 2.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.93 - 8.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.406 + 0.704i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + (0.593 - 1.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.30 - 7.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.00 + 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 - 16.1T + 73T^{2} \)
79 \( 1 + (-6.14 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.36 + 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (3.35 + 5.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.342540279255162568009468230804, −8.518250601936546941235918447061, −7.58429920365149922705541002164, −7.02151384634562822009846919511, −6.57070785057751701848767918644, −5.31751771978489603661101618978, −4.18043644921588831470435546634, −3.32493700342369386432188322078, −2.31885024556162471864056995112, −1.29653954849416531084490055388, 0.76225284782453407957670161942, 2.39478651645325707085980130162, 3.54073004453757510665710034666, 3.98888268478179936792139278375, 5.23335757918541685946464639957, 5.67340719256537401192259357353, 7.00716100143022713037310855276, 7.979596646657312015826589567949, 8.480733997686995913424423463686, 9.203223458450899461328836411534

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