Properties

Label 2-21e2-21.17-c1-0-8
Degree $2$
Conductor $441$
Sign $0.0285 + 0.999i$
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  + (−1.00 + 0.581i)2-s + (−0.322 + 0.559i)4-s − 3.07i·8-s + (−5.68 − 3.28i)11-s + (1.14 + 1.98i)16-s + 7.64·22-s + (−1.65 + 0.957i)23-s + (2.5 − 4.33i)25-s − 8.89i·29-s + (3.02 + 1.74i)32-s + (−5.29 − 9.16i)37-s − 5.29·43-s + (3.67 − 2.12i)44-s + (1.11 − 1.93i)46-s + 5.81i·50-s + ⋯
L(s)  = 1  + (−0.712 + 0.411i)2-s + (−0.161 + 0.279i)4-s − 1.08i·8-s + (−1.71 − 0.990i)11-s + (0.286 + 0.496i)16-s + 1.63·22-s + (−0.345 + 0.199i)23-s + (0.5 − 0.866i)25-s − 1.65i·29-s + (0.534 + 0.308i)32-s + (−0.869 − 1.50i)37-s − 0.806·43-s + (0.553 − 0.319i)44-s + (0.164 − 0.284i)46-s + 0.822i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0285 + 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0285 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277091 - 0.269302i\)
\(L(\frac12)\) \(\approx\) \(0.277091 - 0.269302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (1.00 - 0.581i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.68 + 3.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.65 - 0.957i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.29 + 9.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.357 - 0.206i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.61623725751131083635014041255, −9.987450154654768747252122329125, −8.871981508836654692805418432335, −8.139564421547896584715234188945, −7.54315715504810124046759859896, −6.33978594494924563396115250004, −5.27589340010113478833849481636, −3.91718228606149615804322460251, −2.65377120116169940447090279516, −0.30102433333170685431838345016, 1.69346674897828024174978426636, 2.96150892377427860592915806943, 4.81802203137563566231016444833, 5.39034195192796013663438381860, 6.90174174762214131592634113616, 7.911024739443101628484552087364, 8.714786393466723663044590574970, 9.727277897338701523275218693732, 10.38232985653315038242567726679, 10.97546806425431308011137369702

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