Properties

Label 2-21e2-7.2-c3-0-0
Degree $2$
Conductor $441$
Sign $0.198 + 0.980i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.76 + 3.05i)2-s + (−2.23 − 3.87i)4-s + (−1.03 + 1.79i)5-s − 12.4·8-s + (−3.66 − 6.35i)10-s + (24.5 + 42.5i)11-s − 44.8·13-s + (39.8 − 69.0i)16-s + (13.2 + 22.9i)17-s + (−38.8 + 67.3i)19-s + 9.28·20-s − 173.·22-s + (27.8 − 48.2i)23-s + (60.3 + 104. i)25-s + (79.1 − 137. i)26-s + ⋯
L(s)  = 1  + (−0.624 + 1.08i)2-s + (−0.279 − 0.483i)4-s + (−0.0928 + 0.160i)5-s − 0.551·8-s + (−0.115 − 0.200i)10-s + (0.674 + 1.16i)11-s − 0.956·13-s + (0.623 − 1.07i)16-s + (0.189 + 0.327i)17-s + (−0.469 + 0.812i)19-s + 0.103·20-s − 1.68·22-s + (0.252 − 0.437i)23-s + (0.482 + 0.836i)25-s + (0.597 − 1.03i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1226012749\)
\(L(\frac12)\) \(\approx\) \(0.1226012749\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.76 - 3.05i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (1.03 - 1.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-24.5 - 42.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 44.8T + 2.19e3T^{2} \)
17 \( 1 + (-13.2 - 22.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.8 - 67.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-27.8 + 48.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 121.T + 2.43e4T^{2} \)
31 \( 1 + (152. + 264. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (38.5 - 66.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 248.T + 6.89e4T^{2} \)
43 \( 1 + 147.T + 7.95e4T^{2} \)
47 \( 1 + (-134. + 233. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (70.5 + 122. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (212. + 367. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-293. + 509. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-89.8 - 155. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 674.T + 3.57e5T^{2} \)
73 \( 1 + (118. + 205. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (247. - 429. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 24.4T + 5.71e5T^{2} \)
89 \( 1 + (-536. + 928. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−11.42583188694426371056030723019, −10.09875010754334130337674099374, −9.492481548981025224739485538059, −8.568667240382397362605807858375, −7.54562938941383851362203492380, −7.03760584826129235493746742318, −6.06777989186808597144820333254, −4.93883830061219784387552596452, −3.60026566391656209111885081886, −1.97177779649744958670490088452, 0.05112971679106043644064919278, 1.24810100100459303977182759757, 2.61851756776793985591799656771, 3.59984745803966536901229991466, 5.04296791658268888294204667432, 6.23043155589790646664706362400, 7.33816866176097880217599615778, 8.734941472267164535100204394309, 9.043833870514904533353670649587, 10.14356054494204203916730068583

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