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Properties

Label 2-21e2-63.59-c1-0-27
Degree $2$
Conductor $441$
Sign $0.914 - 0.405i$
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  + (2.05 + 1.18i)2-s + (−0.371 − 1.69i)3-s + (1.81 + 3.14i)4-s + 3.43·5-s + (1.24 − 3.91i)6-s + 3.86i·8-s + (−2.72 + 1.25i)9-s + (7.05 + 4.07i)10-s + 0.313i·11-s + (4.64 − 4.23i)12-s + (−5.09 − 2.94i)13-s + (−1.27 − 5.81i)15-s + (−0.958 + 1.65i)16-s + (−0.476 + 0.825i)17-s + (−7.08 − 0.644i)18-s + (−1.09 + 0.630i)19-s + ⋯
L(s)  = 1  + (1.45 + 0.838i)2-s + (−0.214 − 0.976i)3-s + (0.907 + 1.57i)4-s + 1.53·5-s + (0.507 − 1.59i)6-s + 1.36i·8-s + (−0.907 + 0.419i)9-s + (2.23 + 1.28i)10-s + 0.0946i·11-s + (1.34 − 1.22i)12-s + (−1.41 − 0.816i)13-s + (−0.329 − 1.50i)15-s + (−0.239 + 0.414i)16-s + (−0.115 + 0.200i)17-s + (−1.67 − 0.152i)18-s + (−0.250 + 0.144i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.17027 + 0.672001i\)
\(L(\frac12)\) \(\approx\) \(3.17027 + 0.672001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.371 + 1.69i)T \)
7 \( 1 \)
good2 \( 1 + (-2.05 - 1.18i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
11 \( 1 - 0.313iT - 11T^{2} \)
13 \( 1 + (5.09 + 2.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.476 - 0.825i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.630i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.82iT - 23T^{2} \)
29 \( 1 + (-3.43 + 1.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.53 - 2.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.68 + 4.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0699 + 0.121i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.00 - 1.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 + 5.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.824 - 1.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 + 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.934 - 1.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (0.354 + 0.204i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.23 - 9.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.00 + 6.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.05 - 1.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 6.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.63410144477752128651993313014, −10.36053440405032671715919234881, −9.339484291227544625090586518956, −7.893699241213879320607232367870, −7.14802489610445848091584218343, −6.25338285024147448228827715130, −5.56612488787198719278611468188, −4.93827356471722485008701439019, −3.09879839480041979129226667842, −1.98759854923292715054125046197, 2.08907796493356489408115538822, 2.94152169068105182281067265062, 4.42999125996200712181068785242, 4.98800210236461100465241218472, 5.89927873371801386979957167272, 6.71538966246605785268692036896, 8.818761161148124102865144595718, 9.736588115137957044421215060427, 10.30073540555455384529744635635, 11.08049725158840504376085020006

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