Properties

Label 2-21e2-9.4-c1-0-30
Degree $2$
Conductor $441$
Sign $0.722 + 0.690i$
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  + (−0.0341 − 0.0592i)2-s + (1.69 + 0.355i)3-s + (0.997 − 1.72i)4-s + (1.33 − 2.30i)5-s + (−0.0368 − 0.112i)6-s − 0.273·8-s + (2.74 + 1.20i)9-s − 0.182·10-s + (0.799 + 1.38i)11-s + (2.30 − 2.57i)12-s + (−2.62 + 4.54i)13-s + (3.07 − 3.43i)15-s + (−1.98 − 3.43i)16-s − 6.54·17-s + (−0.0225 − 0.203i)18-s + 1.90·19-s + ⋯
L(s)  = 1  + (−0.0241 − 0.0418i)2-s + (0.978 + 0.205i)3-s + (0.498 − 0.864i)4-s + (0.595 − 1.03i)5-s + (−0.0150 − 0.0459i)6-s − 0.0965·8-s + (0.915 + 0.401i)9-s − 0.0575·10-s + (0.241 + 0.417i)11-s + (0.665 − 0.743i)12-s + (−0.728 + 1.26i)13-s + (0.794 − 0.887i)15-s + (−0.496 − 0.859i)16-s − 1.58·17-s + (−0.00530 − 0.0480i)18-s + 0.436·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.722 + 0.690i$
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: \((2,\ 441,\ (\ :1/2),\ 0.722 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05724 - 0.824913i\)
\(L(\frac12)\) \(\approx\) \(2.05724 - 0.824913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.355i)T \)
7 \( 1 \)
good2 \( 1 + (0.0341 + 0.0592i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.33 + 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.799 - 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.54T + 17T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
23 \( 1 + (-1.53 + 2.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.35 - 5.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.89 + 3.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 2.19T + 73T^{2} \)
79 \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.470T + 89T^{2} \)
97 \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90516858970310816601254447929, −9.674684056818900422512331606517, −9.426543296171161091055704654978, −8.608044647993126843988323816793, −7.24069915460356069516558550976, −6.43228621412110146459335141518, −5.00481274697328436606024748911, −4.37091838195637321145720315633, −2.43106263021967697165113718218, −1.58963681226248689487931915340, 2.24065871890587052691482917074, 2.92955848042129448054602103698, 3.94285167560328458244892788218, 5.76058430310140454098248358587, 7.03958652092189889453729992080, 7.31687864390236402685075544673, 8.484139283662917271394695880728, 9.277291828909427931287232269779, 10.39094411664134690012437632230, 11.10225352618162136777478697846

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