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Properties

Label 2-42e2-12.11-c1-0-68
Degree $2$
Conductor $1764$
Sign $-0.930 + 0.365i$
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  + (−1.35 − 0.405i)2-s + (1.67 + 1.09i)4-s − 3.31i·5-s + (−1.81 − 2.16i)8-s + (−1.34 + 4.48i)10-s − 4.72·11-s + 4.97·13-s + (1.58 + 3.67i)16-s − 0.484i·17-s − 2.29i·19-s + (3.63 − 5.53i)20-s + (6.40 + 1.91i)22-s + 7.97·23-s − 5.97·25-s + (−6.74 − 2.01i)26-s + ⋯
L(s)  = 1  + (−0.958 − 0.286i)2-s + (0.835 + 0.549i)4-s − 1.48i·5-s + (−0.643 − 0.765i)8-s + (−0.424 + 1.41i)10-s − 1.42·11-s + 1.38·13-s + (0.396 + 0.917i)16-s − 0.117i·17-s − 0.525i·19-s + (0.813 − 1.23i)20-s + (1.36 + 0.408i)22-s + 1.66·23-s − 1.19·25-s + (−1.32 − 0.395i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7671990854\)
\(L(\frac12)\) \(\approx\) \(0.7671990854\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.405i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.31iT - 5T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + 0.484iT - 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 - 7.97T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 + 2.39T + 37T^{2} \)
41 \( 1 + 6.55iT - 41T^{2} \)
43 \( 1 + 5.37iT - 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 3.27iT - 67T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 - 2.10T + 73T^{2} \)
79 \( 1 - 12.0iT - 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 + 5.72iT - 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−8.987792067623171346201267652608, −8.312757059556737998369577972255, −7.73629921677996527682927421901, −6.73948263111838195825551722433, −5.64352099497411479548124283629, −4.94143244263856343566343333741, −3.77198966586601541954932193981, −2.65599623532745281161919188110, −1.42699217580657860963683528900, −0.41557429960314004329477087881, 1.46191854332025574474046500738, 2.81627273929989317936858687537, 3.27181555067885910761217738115, 4.98682172946752653427712734974, 5.98386682245498544884720221581, 6.59017420372625959310567306875, 7.33374005287189370215692434167, 8.044052135000426146097679053896, 8.731754872096605258607342332472, 9.742603634433864496263422857809

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