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Properties

Label 2-21e2-9.4-c1-0-16
Degree $2$
Conductor $441$
Sign $-0.904 - 0.426i$
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  + (1.23 + 2.13i)2-s + (1.73 + 0.0789i)3-s + (−2.02 + 3.51i)4-s + (−1.29 + 2.24i)5-s + (1.96 + 3.78i)6-s − 5.05·8-s + (2.98 + 0.273i)9-s − 6.38·10-s + (−2.25 − 3.90i)11-s + (−3.78 + 5.91i)12-s + (0.5 − 0.866i)13-s + (−2.42 + 3.78i)15-s + (−2.16 − 3.74i)16-s + 0.945·17-s + (3.09 + 6.70i)18-s + 4.05·19-s + ⋯
L(s)  = 1  + (0.869 + 1.50i)2-s + (0.998 + 0.0455i)3-s + (−1.01 + 1.75i)4-s + (−0.579 + 1.00i)5-s + (0.800 + 1.54i)6-s − 1.78·8-s + (0.995 + 0.0910i)9-s − 2.01·10-s + (−0.680 − 1.17i)11-s + (−1.09 + 1.70i)12-s + (0.138 − 0.240i)13-s + (−0.625 + 0.977i)15-s + (−0.540 − 0.936i)16-s + 0.229·17-s + (0.729 + 1.57i)18-s + 0.930·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.904 - 0.426i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557773 + 2.49305i\)
\(L(\frac12)\) \(\approx\) \(0.557773 + 2.49305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0789i)T \)
7 \( 1 \)
good2 \( 1 + (-1.23 - 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.29 - 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.945T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.16 + 2.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (3.20 - 5.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.27T + 53T^{2} \)
59 \( 1 + (1.36 - 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.13 + 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 - 1.50T + 73T^{2} \)
79 \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.472 + 0.819i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (5.74 + 9.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.62783807590467910419495333404, −10.60299509757934506711681876132, −9.383435735684464530577169794374, −8.083128643324261488469836137875, −7.903645497640902363766965248304, −6.92830592506145766567673822144, −6.01344381641018650133733615465, −4.84132637571162115249113343366, −3.55713903744432719498868437644, −3.03253701240543513256816078408, 1.31601575231585731486460766373, 2.51461064959421322489554493286, 3.67382935062646437201913784312, 4.52668515029490494701086907385, 5.26602796065890056232445390093, 7.21849051213467371328397707365, 8.205276121267667167765027338139, 9.274138496809550614731158313637, 9.874436438657939238131871876322, 10.82909712009613245853448781677

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