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Properties

Label 2-21e2-7.4-c3-0-32
Degree $2$
Conductor $441$
Sign $0.266 + 0.963i$
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  + (4 − 6.92i)4-s + 70·13-s + (−31.9 − 55.4i)16-s + (28 + 48.4i)19-s + (62.5 − 108. i)25-s + (154 − 266. i)31-s + (−55 − 95.2i)37-s − 520·43-s + (280 − 484. i)52-s + (91 + 157. i)61-s − 511.·64-s + (440 − 762. i)67-s + (595 − 1.03e3i)73-s + 448·76-s + (−442 − 765. i)79-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + 1.49·13-s + (−0.499 − 0.866i)16-s + (0.338 + 0.585i)19-s + (0.5 − 0.866i)25-s + (0.892 − 1.54i)31-s + (−0.244 − 0.423i)37-s − 1.84·43-s + (0.746 − 1.29i)52-s + (0.191 + 0.330i)61-s − 0.999·64-s + (0.802 − 1.38i)67-s + (0.953 − 1.65i)73-s + 0.676·76-s + (−0.629 − 1.09i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.215851846\)
\(L(\frac12)\) \(\approx\) \(2.215851846\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-28 - 48.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + (-154 + 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (55 + 95.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 520T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-91 - 157. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-440 + 762. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + (-595 + 1.03e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (442 + 765. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−10.52443727281703733767782848486, −9.793669872456783475900362390348, −8.734851714721924688401780240230, −7.76970550240274935507859973537, −6.49770407250731333564637511219, −5.98155052175071680745929254055, −4.80332429495097302717263278124, −3.46495679072924515643198194158, −2.01047719789283763296679350191, −0.78048792638021808788462561266, 1.37395620063031139417672714403, 2.93106358904960128310417592302, 3.76920995117335966210653982614, 5.11591640376003062515235918775, 6.44695718963114991338191185798, 7.09584968711626688931932193397, 8.318220335954036810235810003952, 8.767380475425838888391947251652, 10.09418215295189394982834103291, 11.12925556928634260238968211254

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