Properties

Label 2-42e2-7.2-c1-0-12
Degree $2$
Conductor $1764$
Sign $0.701 + 0.712i$
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  − 2·13-s + (4 − 6.92i)19-s + (2.5 + 4.33i)25-s + (−2 − 3.46i)31-s + (5 − 8.66i)37-s + 8·43-s + (7 − 12.1i)61-s + (8 + 13.8i)67-s + (−5 − 8.66i)73-s + (2 − 3.46i)79-s − 14·97-s + (10 − 17.3i)103-s + (−1 − 1.73i)109-s + ⋯
L(s)  = 1  − 0.554·13-s + (0.917 − 1.58i)19-s + (0.5 + 0.866i)25-s + (−0.359 − 0.622i)31-s + (0.821 − 1.42i)37-s + 1.21·43-s + (0.896 − 1.55i)61-s + (0.977 + 1.69i)67-s + (−0.585 − 1.01i)73-s + (0.225 − 0.389i)79-s − 1.42·97-s + (0.985 − 1.70i)103-s + (−0.0957 − 0.165i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589925061\)
\(L(\frac12)\) \(\approx\) \(1.589925061\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.361189171956335614796828198990, −8.472291674775557307128830219238, −7.39745551303209634332365244822, −7.07345698940820125811355153222, −5.88020959733861392847866171818, −5.13500414511092058323493125901, −4.26571316277168790413314224553, −3.14774433695310256136608703272, −2.22379300296538333079826554310, −0.69335406972705442880699528951, 1.14673234245407380756870781757, 2.45455288832451320950951149337, 3.46801658604687274419207993817, 4.45113063728847987930472033292, 5.36215989589383442813981310633, 6.15921684507740690464974393958, 7.07050729325715355674057844797, 7.86147694135727985556909256989, 8.516798118005930908986577574619, 9.530044344664672823687370209582

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