Properties

Label 2-21e2-441.25-c1-0-14
Degree $2$
Conductor $441$
Sign $-0.787 - 0.616i$
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  + (−0.302 + 1.32i)2-s + (−1.73 − 0.0482i)3-s + (0.131 + 0.0633i)4-s + (1.28 + 3.26i)5-s + (0.588 − 2.28i)6-s + (2.63 + 0.220i)7-s + (−1.82 + 2.28i)8-s + (2.99 + 0.167i)9-s + (−4.72 + 0.712i)10-s + (−0.233 + 0.0720i)11-s + (−0.224 − 0.115i)12-s + (5.88 − 1.81i)13-s + (−1.09 + 3.43i)14-s + (−2.06 − 5.72i)15-s + (−2.29 − 2.88i)16-s + (0.132 − 1.76i)17-s + ⋯
L(s)  = 1  + (−0.214 + 0.938i)2-s + (−0.999 − 0.0278i)3-s + (0.0657 + 0.0316i)4-s + (0.573 + 1.46i)5-s + (0.240 − 0.932i)6-s + (0.996 + 0.0832i)7-s + (−0.644 + 0.807i)8-s + (0.998 + 0.0557i)9-s + (−1.49 + 0.225i)10-s + (−0.0703 + 0.0217i)11-s + (−0.0648 − 0.0334i)12-s + (1.63 − 0.503i)13-s + (−0.291 + 0.917i)14-s + (−0.532 − 1.47i)15-s + (−0.574 − 0.720i)16-s + (0.0320 − 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407798 + 1.18255i\)
\(L(\frac12)\) \(\approx\) \(0.407798 + 1.18255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0482i)T \)
7 \( 1 + (-2.63 - 0.220i)T \)
good2 \( 1 + (0.302 - 1.32i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-1.28 - 3.26i)T + (-3.66 + 3.40i)T^{2} \)
11 \( 1 + (0.233 - 0.0720i)T + (9.08 - 6.19i)T^{2} \)
13 \( 1 + (-5.88 + 1.81i)T + (10.7 - 7.32i)T^{2} \)
17 \( 1 + (-0.132 + 1.76i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-1.79 - 3.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.69 - 2.52i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-0.455 + 6.07i)T + (-28.6 - 4.32i)T^{2} \)
31 \( 1 + 0.258T + 31T^{2} \)
37 \( 1 + (6.45 + 4.39i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (1.90 + 0.286i)T + (39.1 + 12.0i)T^{2} \)
43 \( 1 + (8.64 - 1.30i)T + (41.0 - 12.6i)T^{2} \)
47 \( 1 + (0.106 - 0.468i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-8.24 + 5.62i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (2.15 + 2.69i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-13.2 + 6.37i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + (-3.92 - 1.88i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.691 - 0.213i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + 1.81T + 79T^{2} \)
83 \( 1 + (-3.72 - 1.14i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (-7.96 + 7.39i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (5.76 - 9.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.37823360005404725570084288419, −10.70320559544663892244549479335, −9.895760433591198483688876065952, −8.388205697581766522981216881551, −7.57560098997503350300178099713, −6.72375324739870183421046181651, −5.96002885110637709460711968617, −5.38278225445964493473739701492, −3.58770171795969953075598012436, −1.94510476893233791965278682728, 1.09278075635644107885866832655, 1.73078580799559560651069157203, 3.92614196168309369789208454564, 4.97024805913439597331542528374, 5.82492295190740622617456830101, 6.82569165708035058423134183297, 8.444469396750429090906992943178, 9.050445366185948386193772341419, 10.20715098844333473380169718333, 10.80616576869211172844134011203

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