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Properties

Label 2-42e2-21.17-c1-0-10
Degree $2$
Conductor $1764$
Sign $-0.329 + 0.944i$
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  + (−0.382 − 0.662i)5-s + (−1.73 − i)11-s − 0.317i·13-s + (−2.77 + 4.80i)17-s + (3.20 − 1.84i)19-s + (2.74 − 1.58i)23-s + (2.20 − 3.82i)25-s − 6.82i·29-s + (−5.85 − 3.37i)31-s + (0.121 + 0.210i)37-s − 2.74·41-s + 6.82·43-s + (−5.99 − 10.3i)47-s + (−10.6 − 6.12i)53-s + 1.53i·55-s + ⋯
L(s)  = 1  + (−0.171 − 0.296i)5-s + (−0.522 − 0.301i)11-s − 0.0879i·13-s + (−0.672 + 1.16i)17-s + (0.734 − 0.423i)19-s + (0.572 − 0.330i)23-s + (0.441 − 0.764i)25-s − 1.26i·29-s + (−1.05 − 0.606i)31-s + (0.0199 + 0.0345i)37-s − 0.428·41-s + 1.04·43-s + (−0.873 − 1.51i)47-s + (−1.45 − 0.840i)53-s + 0.206i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.047305695\)
\(L(\frac12)\) \(\approx\) \(1.047305695\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 + 0.662i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.317iT - 13T^{2} \)
17 \( 1 + (2.77 - 4.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.20 + 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.74 + 1.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.82iT - 29T^{2} \)
31 \( 1 + (5.85 + 3.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 - 6.82T + 43T^{2} \)
47 \( 1 + (5.99 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.6 + 6.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.62 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.08 + 1.78i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.24 + 3.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.31iT - 71T^{2} \)
73 \( 1 + (10.2 + 5.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.65 - 9.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 + (0.831 + 1.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−8.969036817834444137880198143661, −8.290080949093159545785867663677, −7.60329744753636222523993189001, −6.62010855884428090866871441794, −5.83877697107983525074137773724, −4.92157021796683487279798710996, −4.09557890941787694147523146934, −3.04547312685434154607387101854, −1.93006058262466387980348175304, −0.39638261349962506714130076292, 1.40209614424749763512414852280, 2.76324308867389729291891823287, 3.48635210200897107912420436559, 4.79225115979329295425414082264, 5.30105884270837360639292141784, 6.47057794265037034036214138861, 7.29185385678001499790383224621, 7.71754588039045880476293255416, 8.978588283169814444196317858239, 9.353290232027440590080610845862

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