Properties

Label 2-21e2-49.37-c1-0-18
Degree $2$
Conductor $441$
Sign $0.311 + 0.950i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (1.84 − 1.25i)2-s + (1.09 − 2.78i)4-s + (1.49 + 0.460i)5-s + (1.29 − 2.30i)7-s + (−0.495 − 2.17i)8-s + (3.33 − 1.02i)10-s + (0.278 + 3.71i)11-s + (−0.768 − 0.370i)13-s + (−0.524 − 5.89i)14-s + (0.742 + 0.689i)16-s + (−6.99 − 1.05i)17-s + (0.365 − 0.633i)19-s + (2.91 − 3.65i)20-s + (5.19 + 6.50i)22-s + (−1.11 + 0.168i)23-s + ⋯
L(s)  = 1  + (1.30 − 0.890i)2-s + (0.547 − 1.39i)4-s + (0.667 + 0.205i)5-s + (0.487 − 0.872i)7-s + (−0.175 − 0.767i)8-s + (1.05 − 0.325i)10-s + (0.0839 + 1.11i)11-s + (−0.213 − 0.102i)13-s + (−0.140 − 1.57i)14-s + (0.185 + 0.172i)16-s + (−1.69 − 0.255i)17-s + (0.0839 − 0.145i)19-s + (0.652 − 0.818i)20-s + (1.10 + 1.38i)22-s + (−0.233 + 0.0351i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44667 - 1.77320i\)
\(L(\frac12)\) \(\approx\) \(2.44667 - 1.77320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.29 + 2.30i)T \)
good2 \( 1 + (-1.84 + 1.25i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (-1.49 - 0.460i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.278 - 3.71i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (0.768 + 0.370i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (6.99 + 1.05i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.365 + 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.11 - 0.168i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-2.81 + 3.53i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (3.36 + 5.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.19 - 8.14i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.11 - 9.27i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.48 - 6.52i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (2.66 - 1.81i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-2.22 + 5.65i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (4.93 - 1.52i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-2.49 - 6.36i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-7.43 - 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.24 + 7.83i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.44 + 2.34i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (1.41 - 2.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.8 - 6.64i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.515 + 6.88i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 7.03T + 97T^{2} \)
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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

−11.28836712942498625322875378230, −10.19535153142692028011312948221, −9.690928212937473050534720638465, −8.119497839867770574983787024458, −6.92432681867058827486619751530, −5.99687656323088973525522402948, −4.61892774372509181598210581464, −4.33168103816650415010451369473, −2.70874801827380841405554834547, −1.75002891683150883821716452784, 2.18810547907624731672698846944, 3.61673489067542602720947017384, 4.83103548281003009609490508947, 5.62561187960135650698790389371, 6.25569403423277811628647252076, 7.29483855875109890412049178721, 8.552373672010428252674030926379, 9.139650506038351609642032135568, 10.63113708378729072721155620841, 11.55212372228142707000291477987

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