Properties

Label 2-21e2-441.142-c1-0-43
Degree $2$
Conductor $441$
Sign $-0.0263 + 0.999i$
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  + (0.796 − 2.02i)2-s + (1.67 + 0.428i)3-s + (−2.01 − 1.87i)4-s + (0.0340 − 0.0163i)5-s + (2.20 − 3.06i)6-s + (1.56 + 2.13i)7-s + (−1.47 + 0.712i)8-s + (2.63 + 1.43i)9-s + (−0.00615 − 0.0821i)10-s + (−1.93 − 2.43i)11-s + (−2.58 − 4.00i)12-s + (0.517 − 1.31i)13-s + (5.57 − 1.48i)14-s + (0.0641 − 0.0129i)15-s + (−0.144 − 1.92i)16-s + (−0.288 + 0.267i)17-s + ⋯
L(s)  = 1  + (0.563 − 1.43i)2-s + (0.968 + 0.247i)3-s + (−1.00 − 0.936i)4-s + (0.0152 − 0.00733i)5-s + (0.900 − 1.25i)6-s + (0.592 + 0.805i)7-s + (−0.522 + 0.251i)8-s + (0.877 + 0.478i)9-s + (−0.00194 − 0.0259i)10-s + (−0.584 − 0.732i)11-s + (−0.746 − 1.15i)12-s + (0.143 − 0.365i)13-s + (1.48 − 0.395i)14-s + (0.0165 − 0.00334i)15-s + (−0.0360 − 0.480i)16-s + (−0.0699 + 0.0648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0263 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0263 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0263 + 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0263 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78734 - 1.83512i\)
\(L(\frac12)\) \(\approx\) \(1.78734 - 1.83512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.428i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good2 \( 1 + (-0.796 + 2.02i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-0.0340 + 0.0163i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (1.93 + 2.43i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.517 + 1.31i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.288 - 0.267i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-0.0784 + 0.135i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.328 + 1.43i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (7.05 + 2.17i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (3.54 - 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.77 - 2.09i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.56 - 1.75i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (4.11 - 2.80i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.487 + 1.24i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (7.14 - 2.20i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-11.0 + 7.51i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-3.79 + 3.51i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (2.95 - 5.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.31 - 5.76i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (10.1 + 1.53i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (7.38 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.37 - 11.1i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (5.30 + 13.5i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.43 - 5.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07348624395944260322620471689, −10.13707123650115817268830037997, −9.293792802895468103526255083002, −8.424125755751305364151036627377, −7.52005471454492537937559979975, −5.67303655522255135992077328519, −4.74108768294798531580035758566, −3.57268097373784553700423974214, −2.73645741735287592396861003268, −1.68531069415002286841662423967, 2.00006951746111165966403530380, 3.85375770792242683469022331258, 4.55457919925735772573399842050, 5.77582061816143957477570483926, 7.00586100419673661157967960170, 7.53465537051312894770707503344, 8.156419205958045207164812466685, 9.231022746212528548653448365973, 10.23833350487862926185723435240, 11.41361589518187449373303911544

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