Properties

Label 2-21e2-63.58-c1-0-14
Degree $2$
Conductor $441$
Sign $0.266 - 0.963i$
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  + 1.34·2-s + (1.11 + 1.32i)3-s − 0.184·4-s + (1.26 + 2.19i)5-s + (1.5 + 1.78i)6-s − 2.94·8-s + (−0.520 + 2.95i)9-s + (1.70 + 2.95i)10-s + (−0.233 + 0.405i)11-s + (−0.205 − 0.245i)12-s + (2.91 − 5.04i)13-s + (−1.49 + 4.12i)15-s − 3.59·16-s + (1.93 + 3.35i)17-s + (−0.701 + 3.98i)18-s + (−1.09 + 1.89i)19-s + ⋯
L(s)  = 1  + 0.952·2-s + (0.642 + 0.766i)3-s − 0.0923·4-s + (0.566 + 0.980i)5-s + (0.612 + 0.729i)6-s − 1.04·8-s + (−0.173 + 0.984i)9-s + (0.539 + 0.934i)10-s + (−0.0705 + 0.122i)11-s + (−0.0593 − 0.0707i)12-s + (0.807 − 1.39i)13-s + (−0.387 + 1.06i)15-s − 0.899·16-s + (0.470 + 0.814i)17-s + (−0.165 + 0.938i)18-s + (−0.250 + 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02085 + 1.53737i\)
\(L(\frac12)\) \(\approx\) \(2.02085 + 1.53737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 \)
good2 \( 1 - 1.34T + 2T^{2} \)
5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.233 - 0.405i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.91 + 5.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.93 - 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 - 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0530 - 0.0918i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + (-3.84 + 6.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.11 - 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.33T + 47T^{2} \)
53 \( 1 + (-0.358 - 0.620i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.736T + 59T^{2} \)
61 \( 1 + 0.958T + 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (5.13 + 8.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.05 - 7.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.80 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14607216853367503620505204836, −10.33834159416834409990099913645, −9.753725601176336408088343494095, −8.584736583384662868316029499275, −7.76470272800357099836852892728, −6.13314404400974424362078642016, −5.63737989354562977938657668014, −4.28548439720294619368379428138, −3.41893748267040174245255562083, −2.53500532278611446069526099390, 1.31871481386102142676218959840, 2.84542812396336605116921449482, 4.09226506078601596192302880817, 5.09281052735818715724015341195, 6.11475569511207635359106689895, 7.00164132332798057647708930204, 8.447608252226106449821709354633, 9.000740150035353248886812931328, 9.655926347046870884739240067802, 11.40308433110812593357618808286

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