Properties

Label 2-21e2-63.4-c1-0-26
Degree $2$
Conductor $441$
Sign $0.986 + 0.165i$
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  + (−0.0341 + 0.0592i)2-s + (1.15 − 1.29i)3-s + (0.997 + 1.72i)4-s + 2.66·5-s + (0.0368 + 0.112i)6-s − 0.273·8-s + (−0.329 − 2.98i)9-s + (−0.0910 + 0.157i)10-s − 1.59·11-s + (3.38 + 0.709i)12-s + (2.62 − 4.54i)13-s + (3.07 − 3.43i)15-s + (−1.98 + 3.43i)16-s + (−3.27 + 5.67i)17-s + (0.187 + 0.0824i)18-s + (0.950 + 1.64i)19-s + ⋯
L(s)  = 1  + (−0.0241 + 0.0418i)2-s + (0.667 − 0.744i)3-s + (0.498 + 0.864i)4-s + 1.19·5-s + (0.0150 + 0.0459i)6-s − 0.0965·8-s + (−0.109 − 0.993i)9-s + (−0.0287 + 0.0498i)10-s − 0.482·11-s + (0.976 + 0.204i)12-s + (0.728 − 1.26i)13-s + (0.794 − 0.887i)15-s + (−0.496 + 0.859i)16-s + (−0.793 + 1.37i)17-s + (0.0442 + 0.0194i)18-s + (0.218 + 0.377i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.986 + 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16467 - 0.180600i\)
\(L(\frac12)\) \(\approx\) \(2.16467 - 0.180600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.15 + 1.29i)T \)
7 \( 1 \)
good2 \( 1 + (0.0341 - 0.0592i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.66T + 5T^{2} \)
11 \( 1 + 1.59T + 11T^{2} \)
13 \( 1 + (-2.62 + 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.27 - 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.950 - 1.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.06T + 23T^{2} \)
29 \( 1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 - 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.69 + 6.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.89 + 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.44 - 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.35 + 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (-1.09 + 1.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.406 - 0.704i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.41 - 5.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.235 - 0.407i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.57 + 4.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.03410547439670621617109146774, −10.22488814214812533908696672813, −9.067191765258372254083672893439, −8.216202103682097658530200104441, −7.62856552304374652245704117323, −6.33258487076560844210446527551, −5.85596397613546072436328288730, −3.86008392131490246677613197479, −2.73765343318381396227981359956, −1.76889882020280695006672838607, 1.84245573626473724460177928099, 2.72699202240785047279347868102, 4.44373015861423033397638375316, 5.40210492497143515545257155775, 6.33082221979368955548520956544, 7.37004665521108149539408710651, 8.909867898154392475993499630422, 9.399910967251486647217309827301, 10.10480055468745031191596488636, 10.96150831022923211971938262368

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