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Properties

Label 2-42e2-63.58-c1-0-5
Degree $2$
Conductor $1764$
Sign $0.0433 - 0.999i$
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  + (1.73 + 0.0789i)3-s + (−1.23 − 2.13i)5-s + (2.98 + 0.273i)9-s + (−2.32 + 4.02i)11-s + (−3.55 + 6.15i)13-s + (−1.96 − 3.78i)15-s + (2.25 + 3.90i)17-s + (−2.16 + 3.74i)19-s + (−2.93 − 5.08i)23-s + (−0.527 + 0.912i)25-s + (5.14 + 0.708i)27-s + (3.48 + 6.04i)29-s − 7.38·31-s + (−4.33 + 6.78i)33-s + (0.363 − 0.629i)37-s + ⋯
L(s)  = 1  + (0.998 + 0.0455i)3-s + (−0.550 − 0.952i)5-s + (0.995 + 0.0910i)9-s + (−0.700 + 1.21i)11-s + (−0.985 + 1.70i)13-s + (−0.506 − 0.977i)15-s + (0.547 + 0.948i)17-s + (−0.496 + 0.859i)19-s + (−0.611 − 1.05i)23-s + (−0.105 + 0.182i)25-s + (0.990 + 0.136i)27-s + (0.647 + 1.12i)29-s − 1.32·31-s + (−0.755 + 1.18i)33-s + (0.0597 − 0.103i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0433 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0433 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0433 - 0.999i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.0433 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.546948673\)
\(L(\frac12)\) \(\approx\) \(1.546948673\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0789i)T \)
7 \( 1 \)
good5 \( 1 + (1.23 + 2.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.32 - 4.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.25 - 3.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.16 - 3.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.93 + 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.48 - 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.38T + 31T^{2} \)
37 \( 1 + (-0.363 + 0.629i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 + (2.52 + 4.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.13T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (-2.16 - 3.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 + (0.742 + 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.91 - 8.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.395467123583998855085882748193, −8.645931635573190009789877611807, −8.020181341751075816985557659246, −7.33532519990057667823941278753, −6.52828918252860357474382260859, −5.06172174963500456354273147111, −4.41530156844755086825077920998, −3.83336612247930154915322841243, −2.35291259163629691458704836445, −1.63639639429373277654652417267, 0.48543406131014476757539569819, 2.48428843006997873869944111069, 3.02042751107858000578435070866, 3.69532555717274829540187857271, 5.01303528503524086140536540735, 5.85796751572227284970957318630, 7.08779608658473297557118303492, 7.66912111336512583942090559985, 8.051836001236218396615740211999, 9.072732409629014521454254097828

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