Properties

Label 2-21e2-49.39-c1-0-16
Degree $2$
Conductor $441$
Sign $-0.726 + 0.687i$
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  + (−0.861 − 0.129i)2-s + (−1.18 − 0.365i)4-s + (−0.317 − 4.23i)5-s + (2.28 + 1.33i)7-s + (2.54 + 1.22i)8-s + (−0.276 + 3.68i)10-s + (−0.0107 − 0.0275i)11-s + (1.53 − 1.92i)13-s + (−1.79 − 1.44i)14-s + (0.0179 + 0.0122i)16-s + (−3.29 − 3.05i)17-s + (−2.70 − 4.68i)19-s + (−1.17 + 5.13i)20-s + (0.00573 + 0.0251i)22-s + (−2.28 + 2.11i)23-s + ⋯
L(s)  = 1  + (−0.609 − 0.0918i)2-s + (−0.592 − 0.182i)4-s + (−0.141 − 1.89i)5-s + (0.863 + 0.504i)7-s + (0.899 + 0.433i)8-s + (−0.0874 + 1.16i)10-s + (−0.00325 − 0.00829i)11-s + (0.425 − 0.533i)13-s + (−0.479 − 0.386i)14-s + (0.00448 + 0.00305i)16-s + (−0.799 − 0.741i)17-s + (−0.620 − 1.07i)19-s + (−0.262 + 1.14i)20-s + (0.00122 + 0.00535i)22-s + (−0.475 + 0.441i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.726 + 0.687i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267306 - 0.671190i\)
\(L(\frac12)\) \(\approx\) \(0.267306 - 0.671190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.33i)T \)
good2 \( 1 + (0.861 + 0.129i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (0.317 + 4.23i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (0.0107 + 0.0275i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-1.53 + 1.92i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (3.29 + 3.05i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (2.70 + 4.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.28 - 2.11i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.662 - 2.90i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.83 + 3.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.34 - 0.414i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (9.33 + 4.49i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-7.47 + 3.60i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.583 - 0.0879i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-8.64 - 2.66i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.267 + 3.57i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (12.0 - 3.73i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.28 + 2.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.49 - 6.55i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-8.56 + 1.29i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (4.05 + 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.95 - 2.45i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.65 + 4.20i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 9.08T + 97T^{2} \)
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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

−10.74219942663387725690657447686, −9.525743551486853601955174159091, −8.775439290744103767031154594307, −8.528886887203552761549668798125, −7.52415476164971056613508337365, −5.64890304169116131369161448947, −4.93706695939416142092142051702, −4.24556933579039665728955105367, −1.88739729025936126834464466722, −0.58425659822016528638594571913, 1.95067918762920997522566952737, 3.63424485172204670425416650455, 4.38860531229714677456254017974, 6.15338127303557338488506221448, 6.98947034268421268177423367868, 7.896209839084173483696695693613, 8.535290102448452351556100577805, 9.886946754518182014702675699439, 10.56380107167713904810947033487, 11.05042373799075265106750858623

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