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Properties

Label 2-21e2-49.2-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.968 - 0.250i$
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  + (−2.24 − 0.693i)2-s + (2.92 + 1.99i)4-s + (0.982 − 0.148i)5-s + (−2.07 − 1.64i)7-s + (−2.26 − 2.83i)8-s + (−2.31 − 0.348i)10-s + (−3.21 + 2.98i)11-s + (−0.00383 − 0.0168i)13-s + (3.51 + 5.13i)14-s + (0.530 + 1.35i)16-s + (−0.149 − 2.00i)17-s + (−0.178 + 0.308i)19-s + (3.17 + 1.52i)20-s + (9.30 − 4.48i)22-s + (0.512 − 6.83i)23-s + ⋯
L(s)  = 1  + (−1.59 − 0.490i)2-s + (1.46 + 0.997i)4-s + (0.439 − 0.0662i)5-s + (−0.782 − 0.622i)7-s + (−0.799 − 1.00i)8-s + (−0.731 − 0.110i)10-s + (−0.969 + 0.899i)11-s + (−0.00106 − 0.00466i)13-s + (0.939 + 1.37i)14-s + (0.132 + 0.337i)16-s + (−0.0363 − 0.485i)17-s + (−0.0408 + 0.0707i)19-s + (0.708 + 0.341i)20-s + (1.98 − 0.955i)22-s + (0.106 − 1.42i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0138382 + 0.108851i\)
\(L(\frac12)\) \(\approx\) \(0.0138382 + 0.108851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 1.64i)T \)
good2 \( 1 + (2.24 + 0.693i)T + (1.65 + 1.12i)T^{2} \)
5 \( 1 + (-0.982 + 0.148i)T + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (3.21 - 2.98i)T + (0.822 - 10.9i)T^{2} \)
13 \( 1 + (0.00383 + 0.0168i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.149 + 2.00i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (0.178 - 0.308i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.512 + 6.83i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (2.89 + 1.39i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (2.52 + 4.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.67 - 6.59i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-6.39 - 8.02i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-2.58 + 3.23i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (9.52 + 2.93i)T + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (4.57 + 3.12i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (10.9 + 1.64i)T + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (11.3 - 7.70i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (3.48 + 6.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.82 - 1.84i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-11.2 + 3.47i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (-0.240 + 0.415i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.429 + 1.88i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (6.29 + 5.84i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.31897496511539613080846838276, −9.819210307878087306592846893047, −9.118738100522547104271331056902, −7.960854663994815637422457165934, −7.29660141569330851477200770805, −6.28944735372700255023880647021, −4.73081282910306864968503823225, −3.04023041945650334545847986075, −1.89122386026873615559624638166, −0.10738410977613120632185619472, 1.87890294189834254022013426445, 3.31949135741155827074846599607, 5.53413925097537687839280795119, 6.11651557443785326162340512985, 7.25172450446876025428878173369, 8.073910264134369969992981078074, 9.045130592370195088133057776213, 9.507782962320265530391247334070, 10.52885747567836587388245176466, 11.06010186683962298956146098728

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