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Properties

Label 2-21e2-63.5-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.782 + 0.623i$
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  + 2.57i·2-s + (1.08 + 1.34i)3-s − 4.64·4-s + (1.16 + 2.01i)5-s + (−3.47 + 2.80i)6-s − 6.82i·8-s + (−0.639 + 2.93i)9-s + (−5.20 + 3.00i)10-s + (3.78 + 2.18i)11-s + (−5.04 − 6.26i)12-s + (−1.14 − 0.660i)13-s + (−1.45 + 3.76i)15-s + 8.30·16-s + (−2.89 − 5.01i)17-s + (−7.55 − 1.64i)18-s + (0.584 + 0.337i)19-s + ⋯
L(s)  = 1  + 1.82i·2-s + (0.627 + 0.778i)3-s − 2.32·4-s + (0.521 + 0.903i)5-s + (−1.41 + 1.14i)6-s − 2.41i·8-s + (−0.213 + 0.976i)9-s + (−1.64 + 0.950i)10-s + (1.14 + 0.658i)11-s + (−1.45 − 1.80i)12-s + (−0.317 − 0.183i)13-s + (−0.376 + 0.972i)15-s + 2.07·16-s + (−0.701 − 1.21i)17-s + (−1.78 − 0.388i)18-s + (0.134 + 0.0774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.782 + 0.623i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.782 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529192 - 1.51314i\)
\(L(\frac12)\) \(\approx\) \(0.529192 - 1.51314i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 - 1.34i)T \)
7 \( 1 \)
good2 \( 1 - 2.57iT - 2T^{2} \)
5 \( 1 + (-1.16 - 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.78 - 2.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.14 + 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.89 + 5.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.584 - 0.337i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.81 + 2.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.86 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.01iT - 31T^{2} \)
37 \( 1 + (1.50 - 2.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 - 5.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.493T + 47T^{2} \)
53 \( 1 + (-3.59 + 2.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.31T + 59T^{2} \)
61 \( 1 - 2.05iT - 61T^{2} \)
67 \( 1 + 4.82T + 67T^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + (-13.0 + 7.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (5.32 + 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.66 - 2.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.7 - 7.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.53801536908844275356155000366, −10.27375469875092737568233803031, −9.497999562220367992775798890276, −8.957686901335383321288484084979, −7.85262612812433341185229609048, −6.94498258110021781963671398771, −6.33344170403238744449049773467, −5.01093515458952666590575972015, −4.29867504320968465326343551838, −2.78877401522932671783225585471, 1.07687581189763974456704639087, 1.89236522565300104394524925242, 3.23371329338150981775163145002, 4.19997773051757343314533748238, 5.51731150835286219103228116856, 6.89079308349871855473402528061, 8.630663129476801184014594725768, 8.807411069175104156120340914529, 9.594647408788367828879178600791, 10.73114996025637718901030388652

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