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Properties

Label 2-42e2-9.7-c1-0-25
Degree $2$
Conductor $1764$
Sign $0.992 + 0.120i$
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  + (1.19 − 1.25i)3-s + (1.80 + 3.12i)5-s + (−0.160 − 2.99i)9-s + (3.01 − 5.21i)11-s + (2.55 + 4.42i)13-s + (6.06 + 1.45i)15-s + 0.222·17-s + 3.42·19-s + (0.509 + 0.883i)23-s + (−3.99 + 6.91i)25-s + (−3.95 − 3.36i)27-s + (−2.83 + 4.91i)29-s + (−2.52 − 4.37i)31-s + (−2.97 − 10.0i)33-s − 3.37·37-s + ⋯
L(s)  = 1  + (0.687 − 0.725i)3-s + (0.805 + 1.39i)5-s + (−0.0534 − 0.998i)9-s + (0.908 − 1.57i)11-s + (0.709 + 1.22i)13-s + (1.56 + 0.375i)15-s + 0.0538·17-s + 0.785·19-s + (0.106 + 0.184i)23-s + (−0.798 + 1.38i)25-s + (−0.761 − 0.648i)27-s + (−0.526 + 0.912i)29-s + (−0.453 − 0.784i)31-s + (−0.517 − 1.74i)33-s − 0.554·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.821948687\)
\(L(\frac12)\) \(\approx\) \(2.821948687\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.19 + 1.25i)T \)
7 \( 1 \)
good5 \( 1 + (-1.80 - 3.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.01 + 5.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.55 - 4.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.222T + 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 + (-0.509 - 0.883i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.83 - 4.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.52 + 4.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.37T + 37T^{2} \)
41 \( 1 + (0.0955 + 0.165i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.71 + 2.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.03 - 1.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.30T + 53T^{2} \)
59 \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.891 + 1.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.49 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.89T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + (6.30 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.59 + 6.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + (-2.72 + 4.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.132258670519329784288547329809, −8.677028645020998100672472028314, −7.52549397297208564732140995438, −6.82762908978075395930177420634, −6.28115433846840930961536254781, −5.63865674035046030965074995876, −3.75547609553799210158604266661, −3.33279717451090639923156822626, −2.25845782678114327049965109073, −1.27722846418711508022825087418, 1.24229594931930813316828139037, 2.18982655774756215069501222132, 3.52266846523822441462542170684, 4.40578437576549391318345207390, 5.14288065303358098201374696864, 5.77317691164704674794941954415, 7.05816439351085781415871905623, 8.005270340077541776131971852471, 8.692643195840415046970830037599, 9.376085460987330672404778504886

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