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Properties

Label 2-21e2-21.5-c3-0-3
Degree $2$
Conductor $441$
Sign $-0.708 + 0.706i$
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  + (3.16 + 1.82i)2-s + (2.67 + 4.63i)4-s + (−8.69 + 15.0i)5-s − 9.68i·8-s + (−55.0 + 31.7i)10-s + (−28.2 + 16.2i)11-s + 42.4i·13-s + (39.0 − 67.7i)16-s + (−42.8 − 74.1i)17-s + (59.9 + 34.5i)19-s − 93.0·20-s − 118.·22-s + (−144. − 83.5i)23-s + (−88.8 − 153. i)25-s + (−77.5 + 134. i)26-s + ⋯
L(s)  = 1  + (1.11 + 0.645i)2-s + (0.334 + 0.578i)4-s + (−0.778 + 1.34i)5-s − 0.428i·8-s + (−1.74 + 1.00i)10-s + (−0.773 + 0.446i)11-s + 0.906i·13-s + (0.610 − 1.05i)16-s + (−0.610 − 1.05i)17-s + (0.723 + 0.417i)19-s − 1.04·20-s − 1.15·22-s + (−1.31 − 0.757i)23-s + (−0.710 − 1.23i)25-s + (−0.585 + 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.708 + 0.706i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.708 + 0.706i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7086908687\)
\(L(\frac12)\) \(\approx\) \(0.7086908687\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.16 - 1.82i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (8.69 - 15.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.2 - 16.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 42.4iT - 2.19e3T^{2} \)
17 \( 1 + (42.8 + 74.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-59.9 - 34.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (144. + 83.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 254. iT - 2.43e4T^{2} \)
31 \( 1 + (281. - 162. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (172. - 299. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 182.T + 6.89e4T^{2} \)
43 \( 1 - 140.T + 7.95e4T^{2} \)
47 \( 1 + (-21.5 + 37.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-167. + 96.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (55.6 + 96.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-78.4 - 45.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-524. - 908. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 464. iT - 3.57e5T^{2} \)
73 \( 1 + (158. - 91.3i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-119. + 206. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 375.T + 5.71e5T^{2} \)
89 \( 1 + (719. - 1.24e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 638. iT - 9.12e5T^{2} \)
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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

−11.55960958088886334809030226623, −10.42757599503295755286850477801, −9.668472340887127421753747283273, −8.111850678474520523317606912737, −7.12382093288871429399124563944, −6.76633789777931425786210084560, −5.57635749917794871459167490174, −4.46418769225869275567835430767, −3.61328703635274158393735451251, −2.46492528003024083388208082994, 0.14770430605780650436234951151, 1.79577528869057586540325161693, 3.36045813478626955697025629244, 4.09303391138947032517876704427, 5.22368068950718307812915863759, 5.66691243480177681541762292421, 7.58089002568622192808428578104, 8.296554053961573147701056310309, 9.118924939239614342995127193885, 10.58450163848965360798540719272

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