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Properties

Label 2-21e2-7.4-c3-0-10
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  + (−2.69 − 4.67i)2-s + (−10.5 + 18.3i)4-s + (−7.78 − 13.4i)5-s + 71.0·8-s + (−42.0 + 72.8i)10-s + (−15.9 + 27.6i)11-s − 72.5·13-s + (−107. − 185. i)16-s + (14.5 − 25.1i)17-s + (54.4 + 94.2i)19-s + 329.·20-s + 172.·22-s + (27.6 + 47.8i)23-s + (−58.8 + 101. i)25-s + (195. + 339. i)26-s + ⋯
L(s)  = 1  + (−0.954 − 1.65i)2-s + (−1.32 + 2.29i)4-s + (−0.696 − 1.20i)5-s + 3.14·8-s + (−1.32 + 2.30i)10-s + (−0.438 + 0.758i)11-s − 1.54·13-s + (−1.67 − 2.90i)16-s + (0.207 − 0.358i)17-s + (0.657 + 1.13i)19-s + 3.68·20-s + 1.67·22-s + (0.250 + 0.434i)23-s + (−0.470 + 0.815i)25-s + (1.47 + 2.56i)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: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5879015858\)
\(L(\frac12)\) \(\approx\) \(0.5879015858\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.69 + 4.67i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (7.78 + 13.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15.9 - 27.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 72.5T + 2.19e3T^{2} \)
17 \( 1 + (-14.5 + 25.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-54.4 - 94.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-27.6 - 47.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 17.7T + 2.43e4T^{2} \)
31 \( 1 + (-28.0 + 48.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-147. - 256. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 - 16.8T + 7.95e4T^{2} \)
47 \( 1 + (255. + 443. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (132. - 229. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-127. + 220. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (36.4 + 63.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-253. + 438. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 827.T + 3.57e5T^{2} \)
73 \( 1 + (186. - 322. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (514. + 890. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 453.T + 5.71e5T^{2} \)
89 \( 1 + (166. + 287. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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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

−10.19776483721130727803869669319, −9.756335979204156561904897338299, −8.928918338991264811975601119563, −7.896729524345918714799836023543, −7.50959170776323897827979529508, −5.04452867026313272124053725301, −4.36030321569101895138747947456, −3.10699575284816645120565089034, −1.84632974867831959072875455754, −0.62068699730918042404139044238, 0.50088822982715056053207586885, 2.80516344875748255882998481752, 4.55180340333774183080715752433, 5.61603022020499651169972363983, 6.64956800242247056189517568724, 7.37665500376489974896164782483, 7.85678691581801749557436597564, 8.955804360722440103687366159062, 9.845511727309642020841538861883, 10.66739620406271791469538661223

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