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Properties

Label 2-21e2-9.7-c1-0-31
Degree $2$
Conductor $441$
Sign $-0.965 - 0.259i$
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  + (1.35 − 2.35i)2-s + (−0.521 + 1.65i)3-s + (−2.68 − 4.65i)4-s + (−0.793 − 1.37i)5-s + (3.17 + 3.46i)6-s − 9.15·8-s + (−2.45 − 1.72i)9-s − 4.30·10-s + (0.674 − 1.16i)11-s + (9.08 − 2.00i)12-s + (−1.58 − 2.75i)13-s + (2.68 − 0.593i)15-s + (−7.05 + 12.2i)16-s − 2.80·17-s + (−7.38 + 3.43i)18-s − 0.625·19-s + ⋯
L(s)  = 1  + (0.959 − 1.66i)2-s + (−0.301 + 0.953i)3-s + (−1.34 − 2.32i)4-s + (−0.354 − 0.614i)5-s + (1.29 + 1.41i)6-s − 3.23·8-s + (−0.818 − 0.574i)9-s − 1.36·10-s + (0.203 − 0.352i)11-s + (2.62 − 0.580i)12-s + (−0.440 − 0.763i)13-s + (0.692 − 0.153i)15-s + (−1.76 + 3.05i)16-s − 0.679·17-s + (−1.74 + 0.809i)18-s − 0.143·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.965 - 0.259i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170561 + 1.29101i\)
\(L(\frac12)\) \(\approx\) \(0.170561 + 1.29101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.521 - 1.65i)T \)
7 \( 1 \)
good2 \( 1 + (-1.35 + 2.35i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.793 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.674 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.58 + 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
19 \( 1 + 0.625T + 19T^{2} \)
23 \( 1 + (-0.142 - 0.246i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.27 + 3.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.71 - 6.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.02T + 37T^{2} \)
41 \( 1 + (5.01 + 8.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.57 + 9.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.78T + 53T^{2} \)
59 \( 1 + (-2.28 - 3.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.192 - 0.333i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.26 - 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 - 0.468T + 73T^{2} \)
79 \( 1 + (-7.85 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.99 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 + (7.22 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.62699677626109947716341078875, −10.23868515736110254466376743787, −9.199217615580957028655510653103, −8.464875804420755699981610655344, −6.27561079249535572548047443158, −5.22340207790490517479131373033, −4.56810291679702826599404287404, −3.69004437252429458760127831698, −2.60083622594374914855787070590, −0.62395319353356468149838944658, 2.71980233375980187094577158770, 4.16365650865903958170947666631, 5.12916380844027454837487953043, 6.34322050221983798584913351979, 6.77502010077449873918681302488, 7.54731303875891361570831142882, 8.313156854964127782542239839237, 9.388074593219714604983162935125, 11.16362421562495226703108890274, 11.93551089667090065554324573646

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